Regularity Estimates for Semi-regular Subdivision via ...

Universita’ degli studi di PaviaUniversita’ degli studi di Milano-Bicocca

Dottorato di ricerca in MatematicaXXXI Ciclo

Smoothness Estimates forSemi-regular Subdivision via

Wavelet Tight Frames

Alberto Viscardi

supervised by

Prof. Lucia Romani

Priv.-Dozent Dr.(USA) Maria Charina

Academic Year 2017-2018

Acknowledgments

In geometric orderAn insulated border [...]Growing up, it all seems so one-sided [...]Detached and subdivided

RushSubdivisions

Maybe starting a Ph.D. thesis in mathematics citing a progressive rock song about thepressure that society applies to each individual for fitting into a prescribed lifestyle modelis not really appropriate. Those lines however stuck into my head since I started workingwith subdivision schemes and it is undeniable that they could be used to describe suchobjects! Moreover my Ph.D. programme started and ended in the office 2112 which isthe name of another success of the Canadian band Rush. As a human being, I want tolook at these coincidences as hints I did the correct choice sailing for this adventure, evenif sometimes I felt I was not capable enough for it. Indeed, “human lives are composed[...] like music. Guided by his sense of beauty, an individual transforms a fortuitousoccurrence [...] into a motif [...]. Without realizing it, the individual composes his lifeaccording to the laws of beauty even in times of greatest distress” (M. Kundera, TheUnbearable Lightness of Being). Now that the adventure is approaching its end, it iscorrect and useful to look back, to analyse what was good and what could have beendone differently, maybe better, but most important to say thank you to all the peoplethat, in either big or small way, influenced this journey.

Looking at what I already wrote, it seems natural to start thanking Federico andJacopo, even if we do not see each other so often, for having me introduced to Rush’smusic and Chiara, even if this is not her top contribution to these years, for being thereason why I read Kundera’s book.

In these years I met a lot of wonderful people that enriched me under various aspects,while a other wonderful people I already knew kept having an active part in my life.I was thinking about mentioning each and every name but, after listing all of them,I decided not to write an enormous and sterile list of names. Some readers will thenforgive me if their names are not explicitly written here: I promise they will get mythanks personally.

I start thanking all the professors I had the pleasure to know, for everything theytaught me, directly or indirectly. I owe them a lot for all the things I learnt, even ifone of them is that I will probably never be as good as a mathematician as them. Aspecial thank goes to Prof. Vignati, for the role he played both during my master degreeand the choice to embark on a Ph.D., and to my supervisors, Prof. Charina and Prof.

Romani, for believing in me, for all the immense support I got during the darkest timesof this journey and for all the patience and time they spent on me.

Then I would like to thank the other Ph.D. students and researchers I met for havingshared discussions, cakes, coffees, the joys and the sorrows of this experience. It hasbeen a pleasure to work, to chat, to play and to enjoy life these years with all of them.In particular, I would like to thank David for everything he did for me during my stayingin Vienna and all the nice and interesting people I had the pleasure to meet because ofhim: he surely helped me calling Vienna my second home and this means a lot to me.Another mention goes to Emil, together with Mia: they are the living proof that evenin the “far north” people are not so cold as they are described and that friendship is nota matter of vicinity nor of amount of time, but of intensity.

My family, together with Elena, even with all the problems and hard times, gave methe possibility to try walking this path in the first place and I will always be grateful forthat. Then, I wish to thank all my friends for always being there when I was in needand for their efforts in helping me keeping my pieces together. They always support meand believe in me more than I do.

Even if she will probably never read these lines, I want to thank Alice for being atmy side at the start of this adventure, for the nice time spent together and for havingtaught me that not all the lessons are learnt for free nor the nice way.

To conclude, I want to go back almost at the beginning and thank Chiara again.Whatever will be, I owe her a lot, but this is not the place to say what. This place isonly to say: thank you.

Thank you so much to all of you for everything we shared during these years and foreverything you gave me: I will bring all those pieces with me and I hope to be worthyof them.

Alberto

Contents

Introduction 1

Notation 6

1 Subdivision Schemes 81.1 Regular vs. Semi-Regular . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Computing Moments and Inner Products . . . . . . . . . . . . . . . . . . 25

1.2.1 Moments of Limit Functions . . . . . . . . . . . . . . . . . . . . . 251.2.2 Gramian and Cross-Gramian Matrices . . . . . . . . . . . . . . . 31

2 Wavelet Tight Frames 352.1 The Unitary and Oblique Extension Principles: from Symbols to Matrices 39

2.1.1 A First Example from Cubic B-spline . . . . . . . . . . . . . . . . 502.2 Semi-regular Dubuc-Deslauriers Wavelet Tight Frames . . . . . . . . . . 63

2.2.1 Wavelet Tight Frames Construction . . . . . . . . . . . . . . . . . 662.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 Regularity Analysis via Wavelet Tight Frames 783.1 Characterization of Holder-Zygmund Spaces Br8,8pRq via Tight Frames . 81

3.1.1 The Case r P p0,8qzN . . . . . . . . . . . . . . . . . . . . . . . . 813.1.2 The Case r P N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.2 Approximation of the Optimal Holder-Zygmund Exponent . . . . . . . . 893.2.1 Two Methods for the Estimation of the Optimal Holder-Zygmund

Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.2.2 Computation of Frame Coefficients . . . . . . . . . . . . . . . . . 913.2.3 Numerical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 92

Conclusions 100

Bibliography 101

Index 106

I

Introduction

There are two main protagonists of this thesis: subdivision schemes and wavelet frames.On the one hand, subdivision schemes are iterative methods for generating curves andsurfaces, widely used in CAGD and animation (see e.g. [10, 52, 57, 63]). On theother hand we have wavelets and wavelet frames, from the seminal works of Daubechies,Meyer, Ron, Shen and many others [16, 20, 24, 50, 55, 56], which are families of functionsthat provide methods for useful decompositions of the elements of application relatedfunction spaces such as L2

pRdq. These two topics are deeply related since both methods

are entwined with the concept of refinable functions, i.e. functions rϕkskPZ satisfying arefinement equation of the type

rϕkskPZ “ PTrϕkp2¨qskPZ (0.1)

for some bi-infinite real-valued matrix P. In particular, subdivision schemes generaterefinable functions which build the foundation for wavelet frame constructions.

One of the major open issue in the realm of subdivision nowadays is to understandhow to construct schemes that produce C2 surfaces in settings with arbitrary topology.In particular, the crucial case is when the initial mesh used for the subdivision processfeatures one or more extraordinary vertices, i.e. vertices with a number of incomingedges different from 6 for triangular meshes or 4 for quadrilateral meshes. Since thesubdivision schemes useful for applications are local, the standard approach would beto try to tune locally a known regular scheme around an extraordinary vertex, in such away that the resulting surface is globally C2. Not having a clue on how to do this tuningprocess properly, the consequential step is to go deeper into the smoothness analysisfor more insight. In this direction, results were obtained exploiting spectral analysis,e.g. [52, 54, 62, 63], however without achieving a full understanding of the smoothnessphenomena. The initial idea behind this work was to follow the same direction througha different path, i.e. applying another method for the analysis of the global subdivisionsmoothness. In the regular (shift-invariant) setting, other methods have been usedsuccessfully for this task, such as Fourier techniques, the Joint Spectral Radius approach[6, 25] and wavelet analysis [20, 50]. The first two of them do not naturally extend tothe case of extraordinary vertices since they rely on the shift-invariance of the mesh. Wechoose instead to focus on the latter, the wavelet analysis. To test the availability of thispath, we focused on the easiest non-regular setting, as done in [29], i.e. we consideredunivariate schemes over a semi-regular initial mesh given by

t0 “ ´h`NY t0u Y hrN, h`, hr P p0,8q. (0.2)

1

Introduction

In the regular case, wavelet analysis rely on the characterization of Besov spacesBrp,qpRq provided by Lemarie and Meyer [47] in their follow-up on the results by Frazierand Jawerth [35].

Theorem 0.1 ([50], Section 6.10). Let s ą 0 and 1 ď p, q ď 8. Assume

φk “ φ0p¨ ´ kq : k P Z(

Y

ψj,k “ 2pj´1q{2ψ1,0p2j´1¨ ´kq : k P Z, j P N

(

Ă CspRq

is a compactly supported orthogonal wavelet system with v vanishing moments.Then, for r P p0,minps, vqq,

Brp,qpRq “

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ

bj,kψj,k : takukPZ P `ppZq,

!

2jpr`12´ 1pq }tbj,kukPZ}`p

)

jPNP `qpZq

+

.

To be able to apply Theorem 0.1, i.e. to extract the smoothness of a given function ffrom the decay of its coefficients

tak “ xf, φky : k P Zu and tbj,k “ xf, ψj,ky : j P N, k P Zu, (0.3)

one must first compute these inner products. In the context of subdivision, the analyticexpressions neither of the analysed function f nor of the functions φk, ψj,k are usuallyknown. However, in the regular setting, the desired inner products can be computedexplicitly (or numerically) using results of [44]. Unfortunately, in the semi-regular casethe exact same strategy does not work. The questions naturally arising at this pointare the following. Can we construct function systems that mimic the properties oforthogonal wavelets in the semi-regular setting? Can we characterize at least Br8,8pRqsimilarly to Theorem 0.1? And will we be able to compute the inner products in (0.3)?In this work we are able to answer all these questions affirmatively.

The natural choice to generalize wavelets are wavelet tight frames. These familiesof functions are similar to wavelets since they are also based on refinable functionsΦ “ rφkskPZ and retain the underlying multi-resolution structure, i.e. there exists asequence of bi-infinite matrices tQjujPN such that

Ψj “ 2j{2 QTj Φp2j¨q, j P N. (0.4)

On the other hand wavelet tight frames are more flexible than wavelets, not requiringeither orthogonality or linear independence. The property which is kept is the perfectreconstruction property for L2

pRq, i.e.

f “ÿ

kPZ

xf, φkyφk `ÿ

jPN

ÿ

kPZ

xf, ψj,ky ψj,k, f P L2pRq. (0.5)

The choice of tight frames over more general systems, such as dual frames (see e.g.[23, 30, 31, 37, 38, 42]), is due to the easier applicability of the former which use the samefunctions in (0.5) both for the computation of the coefficients and for the reconstructionof f .

2

Introduction

In the stationary regular setting, i.e. when the subdivision rules are shift-invariant anddo not change between the iterations, the so-called Unitary Extension Principle (UEP)[55, 56] and Oblique Extension Principle (OEP) from [16, 24] are used for constructionsof wavelet tight frames with one or more vanishing moments. UEP and OEP are based onFourier techniques and on factorizations of trigonometric polynomials. A generalizationof the UEP procedure for nonstationary regular schemes, when the subdivision rules canchange from one iteration to the other, was presented in [40]. A general setting that alsocovers the semi-regular case is the one proposed in [14, 15] where matrix formulations ofthe UEP and OEP is given and examples of wavelet tight frames based on non-uniformB-spline schemes are presented.

We construct wavelet tight frames with n vanishing moments from the semi-regularDubuc-Deslauriers 2n-point subdivision schemes and, to meet this goal, we also presentthe convergence analysis of such semi-regular schemes. The family of Dubuc-Deslauriers2n-point schemes was introduced in [27] in the regular case and extended to meshes ofthe type (0.2) in [62]. Our convergence analysis of this family uses the local eigenvalueanalysis [52, 63]. Our construction of the corresponding wavelet tight frames on theregular part of the mesh uses the UEP and is based on the well known (see e.g [51]) linkbetween Dubuc-Deslauriers and Daubechies refinable functions [20]. The interpolationand polynomial generation (up to degree 2n´ 1) properties of the corresponding subdi-vision schemes ensure n vanishing moments for the framelets. On the irregular part ofthe mesh, in a neighborhood of t0p0q, we apply the matrix factorization technique from[14, 15]. Similarly to [8], instead of factorizing a certain global positive semi-definitematrix, we used the regular framelets to reduce the process to the factorization of a fi-nite positive semi-definite matrix involving an appropriate approximation of the inverseGramian matrix, which guarantees n vanishing moments of the framelets. However, theexistence of the underlying refinable functions is ensured only for certain values of hr{h`.

The advantage of our construction is in its simplicity. Indeed, with our UEP basedconstruction we obtain regular framelets with n vanishing moments without endeav-ouring into more tedious computations required in general by the OEP. Furthermore,compared to the B-spline based wavelet tight frames in [16] (the only other semi-regularwavelet tight frame in the literature) whose filters have size (number of non-zero coeffi-cients) 3n ´ 1, the corresponding filters obtained from the Dubuc-Deslauriers 2n-pointframelets are of size 2n` 1. The disadvantage occurs on the irregular part of the mesh,where our filters have possibly larger supports.

Using our wavelet tight frames we are able to estimate the regularity of other semi-regular subdivision schemes. Our method relies on a new characterization of Holder-Zygmund spaces, Br8,8pRq, r ą 0. It generalizes successful wavelet frame methods[9, 10, 11, 20, 43, 50, 53, 58] from the regular to the semi-regular and even to the irregularsetting. In comparison to the method in [21], our approach yields numerical estimates forthe optimal Holder-Zygmund regularity of a refinable function given just the refinementequation without requiring any ad hoc norm estimates for the corresponding subdivisionscheme. Our numerical estimates turn out to be optimal in all considered cases andrequire fewer computational steps than the standard linear regression method.

3

Introduction

We provide a generalization of Theorem 0.1 for function systems

F “

φk : k P Z(

Y

ψj,k : j P N, k P Z(

Ă L2pRq

satisfying properties (3.2)-(3.7). These requirements express conditions about the local-ized properties of the framelets ψj,k. They also guarantee a quasi-uniform behaviour ofthe system over R, even if no shift-invariance is required as for orthogonal wavelets. Ourmain result states the following:

Theorem 0.2. Let s ą 0 and v P N. Assume F Ă CspRq satisfies assumptions (3.2)-(3.7) with v vanishing moments. Then, for r P p0,minps, vqq,

Br8,8pRq “

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ

bj,kψj,k : takukPZ P `8pZq,

!

2jpr`12q }tbj,kukPZ}`8

)

jPNP `8pZq

+

.

The setting described by assumptions (3.2)-(3.7) includes some cases not addressedin the results of Frazier and Jawerth [35] or of Cordero and Grochenig in [18]. Theresults of [35] require that the elements of F in the decomposition of Brp,qpRq are linkedto dyadic intervals. The results in [18] impose the so-called localization property whichimplies that the system F is semi-orthogonal (in particular, non-redundant).

On the other hand, one could view assumptions (3.2)-(3.7) to be somehow restrictive,since they were designed to fit wavelet tight frames F constructed using results of [15],such as the one constructed here from the Dubuc-Deslauriers 2n-point schemes. Forfunction families F satisfying (0.4), assumptions (3.3)-(3.7) reflect properties of thematrices tQjujPN: (3.3) controls the support of the columns of Qj, (3.4) controls theslantedness of Qj and (3.6)-(3.7) are linked to eigenproperties of Qj.

Nevertheless, the spirit of assumptions (3.3)-(3.7) merges with the spirit of atoms andmolecules in [35] and compactly supported orthogonal wavelet systems, for which (3.3)-(3.7) are also satisfied. These similarities are also visible in the structure of the proofsof Propositions 3.5 and 3.6.

For the sake of completeness, we point out that our setting includes some of thewavelet frames considered in [41] for which a characterization of the spaces Br2,2pRq,r P R, is given. However, those frames are shift-invariant, i.e. (0.4) holds with block2-slanted tQjujPN. The approach in [41] applies Fourier techniques that are not feasiblein our case, due to the lack of shift-invariance. The lack of shift-invariance makes alsothe techniques in [3] inapplicable in our case.

To exploit Theorem 0.2 for the smoothness estimates we provide the tools to computethe frame coefficients of the expansion given by the perfect reconstruction property (0.5)for the refinable functions arising from semi-regular subdivision. The main ingredientis the computation of the cross-Gramian matrix between the refinable functions of thewavelet tight frame considered for the analysis and the refinable functions given bythe semi-regular scheme one wants to analyse. To do so we exploit the result in [44](proven in the regular case) to adjust the idea sketched for B-spline in [48] in the case ofgeneric semi-regular schemes. Furthermore, we show how to compute moments of semi-regular refinable functions, thus, extending results in [19] to the semi-regular setting.

4

Introduction

These algorithms are crucial for both the construction of the semi-regular wavelet tightframes and the application of Theorem 0.2 to the smoothness estimates of semi-regularsubdivision.

The organization of the thesis mirrors the steps followed during the three year period ofthe doctoral programme. In Chapter 1, we introduce subdivision schemes, highlighting,in Section 1.1, the similarities and differences between the regular and the semi-regularcase. Then in Section 1.2 we extend the methods in [19, 44, 48] and compute moments(Section 1.2.1) and (cross-)Gramian matrices (Section 1.2.2) in the semi-regular setting.These methods are fundamental for both the construction and the application of ourwavelet tight frames.

Chapter 2 is devoted to the construction of suitable semi-regular wavelet tight frames,see also [60]. We start with the known tools for the construction of wavelet tight framesin the regular case, namely the Unitary and Oblique Extension Principles and theirextension in matrix form to a general formulation that includes the semi-regular setting(Section 2.1). A first detailed example based on cubic B-spline is provided in Section2.1.1. It shows the difficulties arising in the semi-regular case when using the OEP.Next, in Section 2.2, we introduce the family of Dubuc-Deslauriers 2n-point interpolatoryschemes on which our frame construction is based, providing their convergence analysis.Consequently we define the corresponding scaling functions. Our wavelet tight frameconstruction is presented in Section 2.2.1, followed by examples for n “ 1, 2 in Section2.2.2, which illustrate our theoretical results.

At last, in Chapter 3 we discuss the extension of Theorem 0.1 and its application tothe approximation of the smoothness of semi-regular subdivision [7]. In Section 3.1.1,the proof of Theorem 0.2 is split into two cases: Theorem 3.4 treats the case r P p0,8qzNand, in Section 3.1.2, Theorem 3.9 provides the proof of Theorem 0.2 for r P N. Wewould like to emphasize that the results in Sections 3.1.1 and 3.1.2 hold in regular, semi-regular and irregular cases. The proofs in Sections 3.1.1 and 3.1.2 are reminiscent of thecontinuous wavelet transform techniques in [20, 50] and references therein. In Section 3.2,we illustrate our results with several examples. There the underlying semi-regular settingbecomes crucial. In particular, we use wavelet tight frames constructed in Section 2.2, toapproximate the Holder-Zygmund regularity of semi-regular subdivision schemes basedon B-splines, the family of Dubuc-Deslauriers subdivision schemes and interpolatoryradial basis functions based subdivision. Semi-regular B-spline and Dubuc-Deslauriersschemes were introduced, e.g in [21, 62, 63]. The construction of semi-regular RBFsbased schemes is our generalization of [45, 46] to the semi-regular case. The numericalcomputations have been done in MATLAB 2018a on a Windows 10 (x64) laptop (CPU:Intel Core i7-7700HQ 2.80 GHz, RAM: 16 GB).

5

Notation

Vectors and Matrices

We use bold letters and numbers to indicate numerical vectors and matrices, whereaslowercase letters denote vectors and capital letters matrices. Depending on the context,1 can indicate both a vector or a matrix whose components are equal to 1. All vectorsare column vectors and their size, as well as the sizes of matrices, are specified whennot clear from the context. Moreover, sometimes all the finite vectors and matrices areextended to bi-infinite vectors and matrices padded with zeros. For clarity the elementat position p0, 0q is framed in a box.

Vectors and matrices are interpreted as functions of their indices, i.e. vp´3q refersto the component of v at position ´3. We use the MatLab notation for matrices, i.e.Mp´3 : 3, :q indicates the submatrix of M consisting of all the rows with indices between´3 and 3.

The support of a vector v is defined by

supppvq “ ra, bs X Z,

wherea “ maxt k P Z : vpmq “ 0, @m ă k u,

b “ mint k P Z : vpmq “ 0, @m ą k u.

We use the notation rϕkskPZ to indicate the vector of basic limit functions of a conver-gent subdivision scheme and Φ0 “ r φ0,k skPZ the vector of scaling functions obtainedfrom rϕkskPZ after a proper renormalization. These vectors are functions from R to `pZqand all the operations involving them, such as integration or differentiation, are donecomponentwise. Similarly, for two sets of indices or real numbers A and B we define

A ` B “ t a` b : a P A, b P B u.

Function and Sequence Spaces

We use the standard notation for the function spaces CspRq, s P N0, the Holder spaces

CspRq “"

f P C`pRq : supx,hPR

|f p`qpx` hq ´ f p`qpxq|

|h|αă 8

*

,

6

Notation

for s “ ``α, ` P N0, α P p0, 1q, with f p`q denoting the `-th derivative of f , the Zygmundclass

ΛpRq “"

f : RÑ R : supx,hPR

|fpx` hq ´ 2fpxq ` fpx´ hq|

|h|ă 8

*

,

the Lebesgue spaces LppRq, 1 ď p ď 8, and for sequence spaces `ppZq, 1 ď p ď 8.Besov spaces Brp,qpRq, e.g in [50], are defined, for 1 ď p, q ď 8, r P p0,8q, by

Brp,qpRq :“!

f P LppRq : }f}Brp,q “›

›t2jrωrrs`1p pf, 2´jqujPN

›

›

`qă 8

)

,

with the p-th modulus of continuity of order n P N

ωnp pf, xq “ sup|h|ďx

}∆nhpf, ¨q}Lp

and the difference operator of order n P N and step h ą 0

∆nhpf, xq “

nÿ

`“0

ˆ

n

`

˙

p´1qn´`fpx` `hq.

The special case p “ q “ 8 reduces to

Br8,8pRq “

$

&

%

CrpRq X L8pRq, if r P p0,8qzN,

t f P Cr´1pRq X L8pRq : f pr´1q

P ΛpRq u, if r P N.

The corresponding sequence spaces `rp,q, r P p0,8q, are defined, for 1 ď p ď 8 and1 ď q ă 8, by

`rp,q “!

pa, bq P Zˆ pNˆ Zq : }pa, bq}`rp,q “´

}a}q`p `8ÿ

j“1

2jpr`12´ 1pqq}tbj,kukPZ}

q`p

¯1{q)

and, for 1 ď p ď 8 and q “ 8, by

`rp,8 “!

pa, bq P Zˆ pNˆ Zq : }pa, bq}`rp,8 “ max

}a}`p , supjPN

2jpr`12´ 1pq}tbj,kukPZ}`p

(

)

.

7

1 Subdivision Schemes

Subdivision schemes are iterative refining processes that starting from a coarse set ofinitial data (control points) produce functions, curves or surfaces. The local nature ofsubdivision schemes and the simplicity of their implementation made them a standardtool in Computer Aided Geometric Design (CAGD), computer graphics and animation.In this work, we consider only univariate, binary subdivision, i.e. one-dimensional sub-division processes that double the number of control points at each iteration. We focuson the construction of the so-called basic limit functions of such subdivision schemes.The importance of these limit functions is twofold: on one hand they characterize thebehaviour of the scheme, so that one can analyze a scheme via its basic limit functions,while, on the other hand, they are excellent candidates for the construction of wavelettight frames in Chapter 2. In return, wavelet tight frames are the tools we will use forthe analysis of subdivision in Chapter 3.

We start this chapter with a short excursus into subdivision. The selection and orderof facts about subdivision and subdivision properties aims to show the similarities andthe differences between regular and semi-regular settings and at presenting the basictools needed for handling the topics treated in this thesis. For the proofs of well knownresults, the reader is referred to [5, 10, 52, 63].

1.1 Regular vs. Semi-Regular

Both regular and semi-regular subdivision processes we are interested in fall in thefollowing category.

Definition 1.1. Let P be a bi-infinite matrix with compactly supported columns. Itdefines a stationary subdivision operator

P : `pZq2 ÝÑ `pZq2pt, fq ÞÝÑ pu,gq

byup2kq “ tpkq

up2k ` 1q “tpkq ` tpk ` 1q

2

, k P Z (1.1)

andg “ P f . (1.2)

8


Starting with an initial mesh t0 of the form

t0pkq “

$

’

’

’

’

&

’

’

’

’

%

h`k, if k ă 0,

0, if k “ 0,

hrk, if k ą 0,

(1.3)

for some h`, hr P p0,8q, and a vector of initial data f0 P `pZq, the iterative applicationof the subdivision operator P to the couple pt0, f0q, i.e.

ptj, fjq “ P ptj´1, fj´1q “ P jpt0, f0q, j P N,

is called subdivision scheme. The matrix P is called subdivision matrix. If h` “ hr, thenthe mesh t0 is called regular, otherwise it is called semi-regular.

Remark 1.2. The assumption on the compact support of the columns of the subdivisionmatrix P is natural for applications. Moreover, this assumption ensures the localityof the subdivision process, in the sense that each element of f only influences a finitenumber of values of g. 3

Every subdivision scheme satisfying Definition 1.1 over a regular initial mesh t0 iscalled regular. On the other hand, we call a scheme semi-regular if, after a finite numberof subdivision steps, it can be described locally by schemes satisfying Definition 1.1.In general, semi-regular schemes can be non-stationary, i.e. the subdivision matrix Pdepends on j, and be defined over more general initial meshes t0, but, as observed in[62], it suffices to consider the schemes satisfying Definition 1.1 over semi-regular meshest0 to get the full picture.

Remark 1.3. In contrast to a regular mesh, a semi-regular mesh is non-shift-invariant.Indeed, for a regular mesh t0 there exists h ą 0 such that, for every k P Z, t0 ´ hk “

t0p¨ ´ kq. This is not the case for any semi-regular mesh t0 with h` ‰ hr. 3

Once we have the sequence of vector-tuples tptj, fjqujPN, we consider the sequence ofpiecewise linear functions tFjujPN, where Fj interpolate the values fj over the mesh tj,i.e., for x P rtjpkq, tjpk ` 1qs “ 2´jhrk, k ` 1s, k P Z,

Fjpxq “

`

x´ tjpkq˘`

fjpk ` 1q ´ fjpkq˘

`

tjpk ` 1q ´ tjpkq˘ ` fjpkq. (1.4)

The convergence of the subdivision process then is defined as follows.

Definition 1.4. A subdivision scheme satisfying Definition 1.1 is said to be convergentif and only if, for every initial data f0 P `

8pZq, there exists a limit function F P C0

pRqsuch that

limjÑ8

}F ´ Fj}8 “ 0

9


and F ı 0 for at least one set of initial data f0 P `8pZqzt0u. Moreover, if the limit

function F belongs to CrpRq, r ą 0, for any initial data, the scheme is said to be Cr andr is referred as the smoothness of the scheme.

A very basic example of convergent subdivision scheme is the following.

Example 1.5 (Linear B-spline, part I). Consider the subdivision matrix

P “

»

—

—

—

—

—

—

—

—

—

–

1{2. . . 1

1{2 1{2

11{2 1{2

1. . .

1{2

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and an initial regular or semi-regular mesh t0. Then, for f0 P `8pZq, from (1.2) and

(1.4), we have, for every j P N, k P Z,

Fj`

tjp2kq˘

“ fjp2kq “ fj´1pkq “ Fj´1

`

tj´1pkq˘

“ Fj´1

`

tjp2kq˘

and

Fj`

tjp2k ` 1q˘

“ fjp2k ` 1q “fj´1pkq ` fj´1pk ` 1q

2

“Fj´1

`

tj´1pkq˘

` Fj´1

`

tj´1pk ` 1q˘

2“ Fj´1

`

tjp2k ` 1q˘

.

Since Fj and Fj´1 are piecewise linear interpolants on the meshes tj and tj´1, respec-tively, and they coincide on the finer mesh, we have Fj “ Fj´1. Thus, Fj ” F P C0

pRqand the scheme converges. The resulting limit function F belongs to C8pRq whenf0 ” c P R. For general f0 P `

8pRq, however, F is a bounded piecewise linear function

over equispaced knots which is known to be only Lipschitz-continuous, thus, the corre-sponding scheme is C1´ε, ε ą 0. This is the simplest example of a convergent subdivisionscheme called linear B-spline scheme since it produces piecewise linear functions. 4

Convergence is one of the fundamental properties of subdivision. Without convergencesubdivision schemes are very less appealing for visual applications, such as CAGD. More-over, since this work aims to study the smoothness of convergent subdivision, in whatfollows we will often omit the word “convergent” by taking it for granted. When talkingabout graphic design a good trade off between high smoothness and fast implementationis the goal to achieve. The typical aim for the smoothness in applications is C2 smooth-ness. In graphics, for example, the human eye struggles to distinguish C2 curves andsurfaces from the more regular ones. With this link to graphics in mind, it is naturalto refer to j as the resolution level, since all the functions Fj approximate the smooth

10


function F better and better on finer meshes as j grows. This fact is also the reasonwhy they are called subdivision schemes.

Equations (1.1) and (1.2) imply that the schemes we consider are linear. Since theaction of a subdivision operator in Definition 1.1 on the mesh is only a dyadic cut, thesubdivision process is characterized by the subdivision matrix P.

Subdivision Matrix

In the regular case, a subdivision matrix P is a 2-slanted matrix, i.e. there exists acompactly supported vector p such that

Ppk,mq “ ppk ´ 2mq, k,m P Z. (1.5)

Definition 1.6. The vector p in (1.5) is called mask of the scheme defined by P.

Thus, the subdivision step in (1.2) in the regular case coincides with the convolution

gpkq “ÿ

mPZ

ppk ´ 2mq fpmq, k P Z. (1.6)

In the case of a semi-regular scheme instead there exist two compactly supportedvectors p` and pr such that

(i) k`pPq ă krpPq ´ 1, where

k`pPq “ maxpsupppp`qq and krpPq “ minpsupppprqq;

(ii) for every m P Z,

Ppm, kq “

$

&

%

p`p m ` 2 pk`pPq ´ kq q for k ď k`pPq,

prp m ` 2 pkrpPq ´ kq q for k ě krpPq.

(1.7)

The vectors p` and pr are referred, respectively, as left and right regular masks of thescheme defined by P. The submatrix of P given by

Pirr :“ Pp:, k`pPq ` 1 : krpPq ´ 1q, (1.8)

is called irregular part of P.The first difference between the two settings is the structure of the subdivision matrix

P. The semi-regular case allows for a finite number of different consequent columns. Thefirst and the last of such columns, which can be different, are repeated in a 2-slantedfashion as in the regular case. The regular case is a special case of the semi-regular one,where P is 2-slanted and k`pPq, krpPq are chosen such that

suppppq “ t ´krpPq, . . . , ´k`pPq u.

11


Since in Example 1.5 we have a 2-slanted subdivision matrix P that works both in theregular and semi-regular setting, it is natural to ask if this higher generality is necessary.In general, the transition from the regular to the semi-regular setting is not so smooth.Indeed, if F is the limit function of a regular convergent subdivision scheme with thesubdivision matrix P over the regular initial mesh Z which is obtained from the initialvalues f0, then attaching the values Pj f0, j P N, to the refined meshes tj, obtained froma semi-regular initial mesh t0, we obtain the limit function G that satisfies

Gpxq “

$

&

%

F px{h`q, if x ă 0,

F px{hrq, if x ě 0.

Hence, if F P CnpRq, n P N, then, for every k P N, k ď n,

Gpkqpxq “

$

&

%

h´k` F pkqpx{h`q, if x ă 0,

h´kr F pkqpx{hrq, if x ą 0,

(1.9)

and Gpkq is unlikely to be continuous at 0 for h` ‰ hr. To prevent this loss of smoothness,we must allow modifications to the columns of the subdivision matrix that influence thelimit functions at 0.

Usually, dealing with P can be difficult since it is a bi-infinite matrix. Luckily, someproperties can be studied on a finite section of P.

Definition 1.7. The finite square section of the subdivision matrix P defined by

P “ Pp k`pPq : krpPq, k`pPq : krpPq q, (1.10)

is called invariant neighborhood matrix of the scheme.

Remark 1.8. The dimension of the invariant neighbourhood matrix is krpPq´ k`pPq` 1which in the regular case coincides with | suppppq|. Moreover, in the regular case, P doesnot depend on the indexing of the mask p. Indeed, the diagonal of P is always the flippedmask p and the off-diagonal elements are uniquely determined by the 2-slantedness ofP. 3

Via the invariant neighbourhood matrix P we can analyse the right-spectrum of thesubdivision matrix P.

Proposition 1.9 ([63]). Consider a subdivision matrix P and its corresponding in-variant neighborhood matrix P. Then there is a one-to-one correspondence between theright-eigenvectors of P and P. In addition, P and P have the same right-eigenvalues.

Proof. On one hand, due to (1.7), for every k,m P Z such that m ą krpPq and k ď krpPqor m ă k`pPq and k ě k`pPq,

Ppk,mq “ 0. (1.11)

12


Thus, for every bi-infinite vector v,

pP vqpk`pPq : krpPqq “ P vpk`pPq : krpPqq.

In particular, this holds for every right-eigenvector v of P with respect to a right-eigenvalue λ P C, which leads to

P vpk`pPq : krpPqq “ λ vpk`pPq : krpPqq.

On the other hand, with the same argument as in (1.11), P contains all the non-zero di-agonal elements of P. Thus, if v is a right-eigenvector of P with respect to an eigenvalueλ P C, we have that, for

vp1q “

»

—

—

—

–

1

λPpk`pPq ´ 1, k`pPq : krpPqq v

v1

λPpkrpPq ` 1, k`pPq : krpPqq v

fi

ffi

ffi

ffi

fl

,

Ppk`pPq ´ 1 : krpPq ` 1, k`pPq ´ 1 : krpPq ` 1q vp1q “ λvp1q.

Thus, we extended uniquely v to vp1q to a right-eigenvector of Ppk`pPq ´ 1 : krpPq `1, k`pPq´1 : krpPq`1q. This procedure can be repeated indefinitely yielding a bi-infiniteeigenvector of P and, thus, the claim.

Remark 1.10. Since P is a finite matrix, its left and right-eigenvalues are the same. Thisis not true for bi-infinite matrices such as P. Proposition 1.9 gives information onlyabout the right-eigenvalues of P. 3

From the subdivision matrix P of a scheme one can easily read a well known necessarycondition for convergence.

Theorem 1.11 ([5]). Consider a convergent subdivision scheme with subdivision matrixP and let tλiu

Ii“0, I P N, be the right-eigenvalues of P with

|λ0| ě |λ1| ě . . . ě |λI |.

Then,

(i) 1 “ λ0 ą |λ1|;

(ii) P 1 “ 1.

In particular, in the regular case, a consequence of (ii) are the so-called sum rules, i.e.

ÿ

kPZ

pp2kq “ÿ

kPZ

pp2k ` 1q “ 1. (1.12)

In the semi-regular case, (1.7) implies that (1.12) holds for both regular masks p` andpr. This gives a constraint on the support of the columns of Pirr which must be a subset

13


of the sett k`pPq ´ | supppp`q| ` 3, . . . , krpPq ` | supppprq| ´ 3 u.

Example 1.12 (Linear B-spline, part II). From the subdivision matrix P in Example1.5, it is easy to check that the corresponding mask and invariant neighbourhood matrixare respectively

ppkq “

$

&

%

1, if k “ 0,1{2, if k P t´1, 1u,

0, otherwise.and P “

»

–

1{2 1{21

1{2 1{2

fi

fl . (1.13)

The eigenvalues of P are 1 with multiplicity one and 1{2 with multiplicity two, withcorresponding eigenvectors

v1 “

»

–

111

fi

fl , v1{2,1 “

»

–

100

fi

fl and v1{2,2 “

»

–

001

fi

fl .

By Proposition 1.9, P has the same right-eigenvalues of P and the corresponding right-eigenvectors of P can be obtained by extending v1, v1{2,1 and v1{2,2. In particular, theright-eigenvector of P corresponding to the eigenvalue 1 is the bi-infinite vector of allones 1. To extend v1{2,1 to an eigenvector of P, following Proposition 1.9, we want tofind

vp1q “

»

–

av1{2,1

b

fi

fl

such that»

—

—

—

—

–

0 11{2 1{2

11{2 1{2

1 0

fi

ffi

ffi

ffi

ffi

fl

vp1q “1

2vp1q.

This leads to a “ 2 and b “ 0. If we repeat this process indefinitely we get

vp8qpkq “

$

&

%

´k, for k ă 0,

0, for k ě 0.

The extension of v1{2,2 is done in the same way. 4

Basic Limit Functions

As already pointed out, by Definition 1.4, convergent subdivision schemes produce dif-ferent limit functions based on the initial set of data. Among those functions there is afunction family of special interest.

14


Definition 1.13. Consider a convergent subdivision scheme. Its limit functions rϕkskPZ ĂC0pRq, where ϕk is obtained via the subdivision process from the initial data

epkqpmq “ δkm, m P Z,

are called basic limit functions

In the regular case, the global 2-slanted structure of the subdivision matrix P leadsto the following result.

Proposition 1.14 ([5]). Consider a convergent regular subdivision scheme over the mesht0 “ hZ, h ą 0. Let f0, g0 P `pZq such that, for some k˚ P Z, g0pkq “ f0pk´ k

˚q, k P Z.

Then G “ F p¨ ´ hk˚q, where F and G denote the limit function obtained applying thesubdivision scheme to the initial data f0 and g0, respectively.

Proposition 1.14 is a shift-invariance property of regular subdivision, in the sense thatany shift of the initial data results in the shift of the limit functions. In particular, itmeans that the basic limit functions are actually the shifts of a single function, i.e.

ϕkpxq “ ϕ0px´ hkq, x P R, k P Z,

where h is the stepsize of the underlying regular mesh.In the semi-regular case, instead, we only have that

ϕkpxq “

$

&

%

ϕk`pPqp x ` h` pk`pPq ´ kq q for k ď k`pPq

ϕkrpPqp x ` hr pkrpPq ´ kq q for k ě krpPq, x P R, (1.14)

while nothing of the kind can be said about the functions ϕk for k`pPq ă k ă krpPq,leaving us with krpPq ´ k`pPq ` 1 different basic limit functions. This lack of shift-invariance around t0 is one of the most significant difference between the regular andthe semi-regular settings.

Remark 1.15. A global scaling of the initial mesh amounts to a uniform scale of all thebasic limit functions, i.e. if we change the initial mesh from t0 to ht0, for some h ą 0,the basic limit functions will change from ϕk to ϕkp¨{hq, for every k P Z. 3

The importance of the basic limit functions, from the subdivision point of view, residesin the following fundamental result.

Theorem 1.16 ([5]). Consider a convergent subdivision scheme with basic limit func-tions rϕkskPZ. Then, for every initial data f0 P `

8pZq, the corresponding limit function

F satisfiesF pxq “

ÿ

kPZ

f0pkq ϕkpxq, x P R. (1.15)

In particular, the scheme is Cr, r ą 0, if and only if rϕkskPZ Ă CrpRq.

15


The basic limit functions of a scheme have the ability to describe all the limit functionsthat the scheme is able to produce. Thus, the study of a scheme can be reduced to thestudy of its basic limit functions.

From the point of view of the basic limit functions, the necessary condition for theconvergence of a scheme stated in Theorem 1.11, is translated into the following result,which states that the basic limit functions of a convergent scheme form a partition ofunity. In particular, a convergent scheme is able to produce constant polynomials.

Theorem 1.17 ([10]). Consider a convergent subdivision scheme with basic limit func-tions ϕk P C0

pRq. Then, for every x P R,

1T rϕkpxqskPZ “ÿ

kPZ

ϕkpxq “ 1. (1.16)

Definition 1.18. A subdivision scheme on the initial mesh t0, with basic limit functionsrϕkskPZ, is said to generate polynomials of degree n P N0, if for every polynomial π ofdegree at most n there exists a bi-infinite vector c such that

πpxq “ÿ

kPZ

cpkq ϕkpxq “ cT rϕkpxqskPZ, x P R.

If, for every k P Z, cpkq “ πpt0pkqq, then the scheme is said to reproduce polynomialsof degree n.

Refinement Equation

Another fundamental aspect about the basic limit functions of a convergent scheme, isthat they satisfy a very useful equation called refinement equation, which is one of thekey ingredient for our frame construction later on. This is a relation that links the basiclimit functions to their dilated versions.

Theorem 1.19 ([63]). Consider a convergent subdivision scheme with subdivision matrixP and basic limit function rϕkskPZ. Then

rϕkpxqskPZ “ PTrϕkp2xqskPZ, x P R. (1.17)

In particular, in the regular case, over the initial mesh t0 “ hZ, h ą 0,

ϕ0pxq “ÿ

kPZ

ppkq ϕ0p2x´ hkq, x P R, (1.18)

where p is the mask of the scheme.

Example 1.20 (Linear B-spline, part III). Example 1.5 shows that the linear B-splinescheme produces piecewise linear functions interpolating the initial data f P `pZq overt0 “ hZ, h ą 0. Thus, the basic limit function of the scheme is

ϕ0pxq “ p1´ |x|{hqχr´h,hspxq. (1.19)

16


It is easy to see that (1.15) and (1.18) hold. In particular, see Figure 1.1,

ϕ0pxq “1

2ϕ0p2x` hq ` ϕ0p2xq `

1

2ϕ0p2x´ hq, x P R. (1.20)

If we consider the semi-regular initial mesh t0 as in (1.3), with h`, hr ą 0, then the

-1 -0.5 0 0.5 10

0.25

0.5

0.75

1

-1 -0.5 0 0.5 10

0.25

0.5

0.75

1

Figure 1.1: The basic limit function ϕ0 of the linear B-spline scheme on the mesh t0 “ Zon the left and its decomposition as in the refinement equation (1.20), withh “ 1.

corresponding basic limit functions are

ϕkpxq “

$

’

’

’

’

&

’

’

’

’

%

p1´ |x´ h`k|{h`q χh`rk´1,k`1spxq, if k ă 0,

p1` x{h`q χr´h`,0spxq ` p1´ x{hrq χr0,hrspxq, if k “ 0,

p1´ |x´ hrk|{hrq χhrrk´1,k`1spxq, if k ą 0,

x P R. (1.21)

Thus, Proposition 1.14 no longer holds and we have three different basic limit functionsinstead of one (see e.g. Figure 1.2 with h` “ 1 and hr “ 2). Here the refinement equationholds in its matrix form (1.17) but not as in (1.18). Again the basic limit functions forma partition of unity (1.16) and the scheme reproduces polynomials of degree 1. Sincethe subdivision matrix P is the same as in the regular case, see Example 1.5, all theeigenproperties of P are kept. Moreover, we observe that at 0 the basic limit functionsare still C1´ε, ε ą 0. 4

Now that we introduced all the necessary tools, we would like to take a step backand prove Theorem 1.11 to point out a specific fact about the invariant neighbourhoodmatrix and the structure of semi-regular subdivision matrices.

Proof of Theorem 1.11. For the sake of simplicity we only consider the regular case.Let suppppq “ ta, . . . , bu, and, for k P Z, tξ

pkqj ujPN the piecewise linear functions

that interpolate Pj epkq, epkqpmq “ δkm, over the mesh tj “ 2´jt0 approximating ϕk.

We consider the invariant neighborhood matrix P. By Proposition 1.9, studying theright-spectrum of P and the spectrum of P is equivalent. Due to Definition 1.7,

”

ξpkqj ptjpmqq

ı

m“´b,...,´a“ Pj epkqp´b : ´aq, j P N, k P Z. (1.22)

17


-2 -1 0 2 40

0.25

0.5

0.75

1

Figure 1.2: The basic limit functions ϕ´1, ϕ0 and ϕ1 of the linear B-spline scheme onthe semi-regular mesh with h` “ 1 and hr “ 2.

Since the scheme is convergent, the left-hand side of (1.22) converges, for j Ñ 8, toϕkp0q1, for every k P Z. Now, if we suppose |λ0| ă 1, (1.22) implies ϕ0p´hkq “ ϕkp0q “ 0for every k P Z. But in this case, from the refinement equation (1.18) we also get, forx “ hm{2, m P Z,

ϕ0phm{2q “ÿ

kPZ

ppkq ϕ0ppm´ kqhq “ 0.

Repeating this argument it is easy to check that ϕ0pxq “ 0 for every x P t2´jhkujPN,kPZwhich is a dense set in R. At this point, the continuity of ϕ0 implies ϕ0 ” 0 which isagainst the hypotesis of convergence. On the other hand, if we suppose |λ0| ą 1, witheigenvector v0 P Rbá`1

zt0u, we have

8 “ limjÑ8

›

›

›Pj v

›

›

›

8“ lim

jÑ8

›

›

›

›

›

Pj

˜

bá`1ÿ

k“1

vpkq epk´b´1q

¸›

›

›

›

›

8

“

›

›

›

›

›

bá`1ÿ

k“1

vpkq ϕk´b´1p0q 1

›

›

›

›

›

8

ă 8,

which is a contradiction. The only case left then is |λ0| “ 1. Supposing λ0 ‰ 1 impliesthat the limit for j Ñ 8 of λj0 does not exist which means that, for v0 P Rbá`1

zt0u aneigenvector associated to λ0, also the limit of Pj v “ λj0 v does not exist and this isagain in contrast with

limjÑ8

Pj v “

bá`1ÿ

k“1

vpkq ϕk´b´1p0q 1. (1.23)

This leaves us with λ0 “ 1 and the same argument as (1.23) shows that it can onlyhave algebraic multiplicity one with associated eigenspace generated by 1. In particular,

18


since P contains all the non-zero elements of the rows ´b, . . . ,á of P (1.11), P 1 “ 1implies (ii). In particular, from (1.5),

1 “ pP 1qpmq “ÿ

kPZ

ppm´ 2kq “ÿ

k”m mod 2

ppkq, m P Z.

From (1.22) and its limit for j Ñ 8 we see how P encodes the information aboutthe basic limit functions around 0. Indeed, the only basic limit functions that can notvanish at 0 are the one with index k between 1 ´ b and á ´ 1, since the supports ofϕá and ϕ´b start and end at 0 respectively. As for Pirr, due to (1.8) and (1.10), itscolumns are exactly the ones in common with P which refer to the basic limit functionshaving 0 inside their support.

We now focus on the regular case, where we exploit the refinement equation to evaluatethe basic limit function ϕ0.

Proposition 1.21 ([5]). Let ϕ0 be the basic limit function of a regular convergent sub-division scheme over the initial mesh t0 “ hZ, h ą 0, with the subdivision matrix P andthe mask p, suppppq “ ta, . . . , bu. Then,

rϕ0p´hkqskPZ “ PTrϕ0p´hkqskPZ.

In particular, ϕ0p´hkq “ 0 for k ď ´b or k ě á, and

rϕ0p´hkqsk“´b,...,á “ PTrϕ0p´hkqsk“´b,...,á with

bÿ

k“a

ϕ0phkq “ 1. (1.24)

Moreover, for every j P N, m P Z,

rϕ0p2´jhpm´ 2jkqqskPZ “ pPT

qjrϕ0phpm´ kqqskPZ. (1.25)

We can get more insight about ϕ0 looking at the refinement equation (1.18) againfrom a different angle. The structure of (1.18) suggests to pass to the Fourier side.

Theorem 1.22 ([10]). Let p P `pZq be a compactly supported mask. If there existsϕ0 P L

1pRq satisfying the refinement equation associated to p (1.18), for some h ą 0,

thenpϕ0pωq “ pphω{2q pϕ0pω{2q, ω P R,

where pϕ0 is the Fourier transform of pϕ0, i.e.

pϕ0pωq “

ż

Rϕ0pxq e

´2πixω dx, ω P R,

19


and ppωq is the trigonometric polynomial

ppωq :“1

2

ÿ

kPZ

ppkq e´2πikω. (1.26)

In particular, if p is the mask of a convergent subdivision scheme, we have

ppωq “1` e´2πiω

2pr1spωq, (1.27)

where pr1spωq is a trigonometric polynomial with pr1sp0q “ 1.

Definition 1.23. Consider a regular subdivision scheme with compactly supported maskp. The trigonometric polynomial ppωq associated with p in (1.26) is called symbol ofthe scheme.

Remark 1.24. A consequence of (1.5) and Proposition 1.14 is that, given a mask p, allthe subdivision schemes associated to shifts of p are equivalent. This implies that thesymbol ppωq of a scheme is unique up to a factor e´2πikω, k P Z. 3

Every manipulation requiring the symbol of a scheme can only be exploited in theregular case, due to the shift-invariance of the basic limit functions. In the semi-regularcase, the lack of shift-invariance does not allow for a meaningful extension of the conceptof symbol.

Example 1.25 (Linear B-spline, part IV). From (1.26) one has

ppωq “1

4e´2πiω

`1

2`

1

4e2πiω

“1` e´2πiω

2

1` e2πiω

2“

1` e´2πiω

2pr1spωq

as in Theorem 1.22, but also

ppωq “

ˆ

eπiω ` e´πiω

2

˙2

“ pcospπωqq2, ω P R.

In particular, the symbol of the linear B-spline scheme has a double zero at ω “ 1{2.Moreover, from the analytic expression of ϕ0 (1.19), it is easy to check

ÿ

kPZ

ϕ0px´ hkq “ 1 andÿ

kPZ

hk ϕ0px´ hkq “ x, x P R,

which means, by the linearity of the subdivision process, that the linear B-spline schemereproduces polynomials of degree 1. 4Remark 1.26. The multiplicity of the zero at ω “ 1{2 of the symbol ppωq and the degreeof the polynomial generation of the associated scheme are intrinsically linked. Indeed, aregular scheme generates polynomials of degree n P N0 if and only if

ppωq “

ˆ

1` e´2πiω

2

˙n`1

prnspωq,

20


where prnspωq is a trigonometric polynomial such that prnsp0q “ 1 (see e.g. [10]). 3

Via symbol manipulations one can also produce new subdivision schemes from theknown ones.

Proposition 1.27 ([10]). Consider two convergent regular subdivision schemes over thesame regular mesh t0 “ hZ, h ą 0. If ϕ0 P CmpRq, m ą 0, and ζ0 P CnpRq, n ą 0,are the basic limit functions of the two schemes associated to the symbols ppωq and zpωqwith compactly supported masks p and z, respectively, then the symbol

gpωq “ ppωq zpωq, ω P R, (1.28)

defines a convergent subdivision scheme over the same initial mesh t0 with basic limitfunction given by

γ0pxq “

ż

Rϕ0pyq ζ0px´ yq dy, x P R.

Moreover, γ0 P Cmaxpm,nqpRq.

Remark 1.28. In terms of the masks, equation (1.28) for gpωq “ÿ

kPZ

gpkqe´2πikω is a

convolutiongpkq “

ÿ

mPZ

ppmq zpk ´mq, k P Z.

3

Remark 1.29. Due to the smoothing property of the convolution, Proposition 1.27 workseven if one of the functions is a non-continuous solution of a refinement equation, e.g.the indicator function χr0,hs R C0

pRq, h ą 0, which satisfies

χr0,hspxq “ χr0,h{2spxq ` χrh{2,hspxq “ χr0,hsp2xq ` χr0,hsp2x´ hq, x P R,

and, on the Fourier side,

pχr0,hspωq “ pphω{2q pχr0,hspω{2q with ppωq “1` e´2πiω

2, ω P R.

3

Remark 1.30. The basic limit function γ0 in Proposition 1.27 has a wider support thaneach of ϕ0 or ζ0. Indeed, if

suppppq “ t a, . . . , b u and supppzq “ t c, . . . , d u,

then supppgq “ t a` c, . . . , b` d u. By Theorem 1.16,

| supppγ0q| “ hp| supppgq| ´ 1q “ hpb` d´ a´ cq “ | supppϕ0q| ` | supppζ0q|.

3

We illustrate the result of Proposition 1.27 on the following example.

21


Example 1.31 (Cubic B-spline, part I). We choose the symbols in Proposition 1.27 tobe both the one associated to the linear B-spline scheme in Examples 1.5, 1.12, 1.20 and1.25. Then, by Proposition 1.27, we obtain the product symbol

ppωq “

ˆ

1

4e´2πiω

`1

2`

1

4e2πiω

˙2

“

ˆ

1

16e´4πiω

`1

4e´2πiω

`3

8`

1

4e2πiω

`1

16e4πiω

˙

“

ˆ

1` e´2πiω

2

˙4

e4πiω“ pcospπωqq4, ω P R.

Thus, we have

P “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

1{81{2

. . . 3{4 1{81{2 1{2

1{8 3/4 1{8

1{2 1{2

1{8 3{4. . .

1{21{8

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

and P “

»

—

—

—

—

–

1{8 3{4 1{81{2 1{21{8 3{4 1{8

1{2 1{21{8 3{4 1{8

fi

ffi

ffi

ffi

ffi

fl

. (1.29)

The eigenvalues of P are 1, 1{2, 1{4, 1{8 and 1{8 with corresponding eigenvectors

v1 “

»

—

—

—

—

–

11111

fi

ffi

ffi

ffi

ffi

fl

, v1{2 “

»

—

—

—

—

–

´1´1{2

01{2

1

fi

ffi

ffi

ffi

ffi

fl

, v1{4 “

»

—

—

—

—

–

12{11´1{11

2{111

fi

ffi

ffi

ffi

ffi

fl

, v1{8,1 “

»

—

—

—

—

–

10000

fi

ffi

ffi

ffi

ffi

fl

, v1{8,2 “

»

—

—

—

—

–

00001

fi

ffi

ffi

ffi

ffi

fl

.

These vectors can be extended uniquely to eigenvectors of P as in Proposition 1.9, e.g.for v1{2,

vp1q1{2 “

»

–

av1{2

b

fi

fl

22


such that»

—

—

—

—

—

—

—

—

–

0 1{2 1{21{8 3{4 1{8

1{2 1{21{8 3{4 1{8

1{2 1{21{8 3{4 1{8

1{2 1{2 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

vp1q1{2 “

1

2vp1q1{2,

which leads to a “ ´3{2 and b “ 3{2. We can then repeat the process to add a further

component on the top and on the bottom of vp1q1{2, considering the square submatrix of

P obtained by P this time adding two rows and two columns on each sides.The basic limit function ϕ0 (Figure 1.3 for the initial mesh Z), is the convolution of

the hat function (1.19) with itself and is a piecewise cubic polynomial. Moreover, thescheme is C3´ε

pRq, ε ą 0, and generates cubic polynomials. 4

-2 -1 0 1 20

0.25

0.5

0.75

Figure 1.3: The basic limit function ϕ0 of the regular cubic B-spline scheme over theinitial mesh t0 “ Z.

Before we proceed with some computational aspect about subdivision, it is worthwhileto fully review the cubic B-spline scheme of Example 1.31 in a semi-regular setting.

Example 1.32 (Cubic B-spline, part II). Consider the cubic B-spline scheme, Example1.31. The regular basic limit function in Figure 1.3 is a piecewise cubic polynomial withC2 junctions and it is increasing (decreasing) at x “ ´1 (x “ 1). Thus, if we use theregular subdivision matrix (1.29) over a semi-regular mesh with h` ‰ hr, due to (1.9),we end up with ϕ´1 and ϕ1 to be not C1. However, given a semi-regular mesh t0 one cancompute via knot insertion, e.g. with the Oslo algorithm [17], the matrix that describesthe cubic B-splines on t0 as a linear combination of the cubic B-splines on t1. This isindeed the semi-regular subdivision matrix for the cubic B-spline scheme. For example,

23


if we consider the initial semi-regular mesh with h` “ 1 and hr “ 2, we obtain

P “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

1{81{23{4 1{81{2 1{2

. . . 1{8 3{4 1{81{2 1{21{8 25{32 3{32

5{8 3{8

5{24 29/40 1{15

3{5 2{53{20 29{40 1{8

1{2 1{2

1{8 3{4 1{8 . . .

1{2 1{21{8 3{4

1{21{8

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

with the corresponding invariant neighbourhood matrix

P “

»

—

—

—

—

–

1{8 25{32 3{325{8 3{8

5{24 29{40 1{153{5 2{5

3{20 29{40 1{8

fi

ffi

ffi

ffi

ffi

fl

.

The subdivision matrix P has now three irregular columns which correspond to the threeirregular basic limit functions that have 0 in the interior of their supports. Figure 1.4shows the five basic limit functions ϕ´2,...,ϕ2 around 0, the first and the last ones beingthe regular ones, with ϕ2 “ ϕ´2p¨{2 ´ 4q, while ϕ´1, ϕ0 and ϕ1 are the irregular ones,which are not shifts of any other basic limit functions. The eigenvalues of P are 1, 1{2,1{4, 1{8 and 1{8, the same as in the regular case (Example 1.31), but with differentcorresponding eigenvectors

v1 “

»

—

—

—

—

–

11111

fi

ffi

ffi

ffi

ffi

fl

, v1{2 “

»

—

—

—

—

–

´1{2´1{41{121{2

1

fi

ffi

ffi

ffi

ffi

fl

, v1{4 “

»

—

—

—

—

–

1{41{22

´1{222{11

1

fi

ffi

ffi

ffi

ffi

fl

, v1{8,1 “

»

—

—

—

—

–

10000

fi

ffi

ffi

ffi

ffi

fl

, v1{8,2 “

»

—

—

—

—

–

00001

fi

ffi

ffi

ffi

ffi

fl

.

4

24


-4 -3 -2 -1 0 2 4 6 80

0.25

0.5

0.75

Figure 1.4: The basic limit functions ϕ´2,...,ϕ2 of the semi-regular cubic B-spline schemeover the initial semi-regular mesh t0 with h` “ 1 and hr “ 2.

1.2 Computing Moments and Inner Products

In this section, we present algorithms for the computation of two fundamental quantitiesfor the construction and for the application of wavelet tight frames. These two quantitiesare the moments and the (cross-)Gramian matrices.

Definition 1.33. Let f : RÑ R. The quantity

ż

Rxα fpxq dx, α P N0 (1.30)

is called the pα ` 1q-th moment of f .

Definition 1.34. Consider two families of functions F “ rfkskPZ and G “ rgkskPZ. Thematrix

G :“

„ż

Rfkpxq gmpxq dx

k,mPZ(1.31)

is called the cross-Gramian matrix between F and G. If the two families coincide, G iscalled the Gramian matrix of F .

In this section, the families F and G of basic limit functions of subdivision schemes arebuilt from both regular and semi-regular subdivision. We start with the computationof the moments, using the work of Dahmen and Micchelli [19] for the regular case anda simple generalization of their argument to deal with the semi-regular one. Then weproceed by computing inner products between refinable functions. For the regular casewe use an idea of Kunoth [44], while for the semi-regular case we exploit a methodsuggested by Lounsbery [48], proving its viability.

1.2.1 Moments of Limit Functions

We start with the simplest case. Let ϕ0 be the basic limit function of a convergentregular subdivision scheme over the initial mesh t0 “ hZ, h ą 0. In particular it is

25


continuous, compactly supported and it satisfies the refinement equation (1.18) withrespect to a compactly supported mask p P `pZq that satisfies the sum rule (1.12).

Proposition 1.35 ([19]). For every α P N, we have

ż

Rxα ϕ0pxq dx “

1

2p2α ´ 1q

ÿ

0ďβăα

cαpβq

ż

Rxβ ϕ0pxq dx,

where

cαpβq :“

ˆ

α

β

˙

ÿ

kPZ

ppkq phkqα´β, 0 ď β ă α.

Proof. Using the refinement equation (1.18) and the fact that the sequence p is com-pactly supported we have

ż

Rxα ϕ0pxq dx “

ÿ

kPZ

ppkq

ż

Rxα ϕ0p2x´ hkq dx.

After the substitution y “ 2x´ hk, using the binomial formula we get

ż

Rxα ϕ0pxq dx “

1

21`α

ÿ

kPZ

ppkq

ż

Rpy ` hkqα ϕ0pyq dy

“1

21`α

ÿ

kPZ

ppkq

ż

Rϕ0pyq

ÿ

0ďβďα

ˆ

α

β

˙

yβ phkqα´β dy

Now, changing the order of the sums and the integral and recalling the definition ofcαpβq, we arrive at

ż

Rxα ϕ0pxq dx “

1

21`α

ÿ

0ďβďα

cαpβq

ż

Ryβ ϕ0pyq dy

Since, by (1.12),

cαpαq “ÿ

kPZ

ppkq “ 2,

we can bring all the term with β “ α on the left-hand side and get

ˆ

1 ´1

2α

˙ż

Rxα ϕ0pxq dx “

1

21`α

ÿ

0ďβăα

cαpβq

ż

Rxβ ϕ0pxq dx.

Thus the claim follows.

Remark 1.36. From the moments of ϕ0, one can easily compute the moments of its shifts.

26


Indeed, for every α P N0, y P R,

ż

Rxα ϕ0px´ yq dx “

ż

Rpx` yqα ϕ0pxq dx “

ÿ

0ďβďα

ˆ

α

β

˙

yα´βż

Rxβ ϕ0pxq dx.

In particular we haveż

Rϕkpxq dx “

ż

Rϕ0pxq dx

andż

Rxα ϕkpxq dx “

ÿ

0ďβďα

ˆ

α

β

˙

phkqα´βż

Rxβ ϕ0pxq dx, k P Z.

Moreover, for λ ą 0,

ż

Rxα ϕkpλxq dx “

1

λ1`α

ż

Rxα ϕkpxq dx (1.32)

3

Proposition 1.35, together with Remark 1.36, tells us that, if we know the value ofthe integral of ϕ0, all the moments of all the basic limit functions can be computed ina recursive fashion. However, as we already observed in Remark 1.15, the refinementequation (1.18) alone does not give any information about the integral of ϕ0. With thefollowing result we compute the integral of all compactly supported limit functions thata regular scheme can generate. As a consequence, we have that the integral of ϕ0 is veryeasy to compute and depends only on the stepsize h of the initial mesh t0.

Proposition 1.37. Consider a regular convergent subdivision scheme over the initialmesh t0 “ hZ, h ą 0, with mask p P `pZq, subdivision matrix P and basic limit functionϕ0. If f P C0

pRq is the limit function obtained by the subdivision scheme starting withthe compactly supported data f0 P `

8pZq, then

ż

Rfpxq dx “ h

ÿ

kPZ

f0pkq. (1.33)

In particular,ż

Rϕ0pxq dx “ h. (1.34)

Proof. Let tfjujPN be the piecewise linear functions interpolating the data fj “ Pj f0

over the mesh tj “ 2´jt0 “ 2´jhZ. Due to the convergence of the scheme we havelimjÑ8

}f ´ fj}8 “ 0. Now, for every j P N, being fj piecewise linear over the mesh tj, we

27


haveż

Rfjpxq dx “

ÿ

kPZ

fjptjpkqq ` fjptjpk ` 1qq

2ptjpk ` 1q ´ tjpkqq

“h

21´j

ÿ

kPZ

fjpkq ` fjpk ` 1q “h

2´j

ÿ

kPZ

fjpkq

“h

2´j

ÿ

kPZ

ÿ

mPZ

ppk ´ 2mqfj´1pmq “h

21´j

ÿ

mPZ

fj´1pmq

“ hÿ

kPZ

f0pkq ă 8,

where we used (1.6) and (1.12). Thus, by uniform convergence, we have

ż

Rfpxq dx “ lim

jÑ8

ż

Rfjpxq dx “ h

ÿ

kPZ

f0pkq.

In particular, since ϕ0 is obtained by the initial data ep0q “ rδ0kskPZ, we get (1.34).

Remark 1.38. If one is able to compute every moment of every basic limit function ϕk,k P Z, then one can compute all the inner products between every polynomial π andevery limit function f . Indeed, if

πpxq “Nÿ

α“0

πα xα and fpxq “

ÿ

kPZ

f0pkq ϕkpxq,

for some f P `pZq, then

x π, f y “

ż

R

Nÿ

α“0

πα xαÿ

kPZ

f0pkq ϕkpxq dx “ÿ

kPZ

f0pkqNÿ

α“0

πα

ż

Rxα ϕkpxq dx.

3

Remark 1.39. Proposition 1.35 can be generalized in a straightforward way to higherdimensional refinable functions with general diagonal dilation, i.e. if ϕ0 P C0

pRdq, d P N,

withϕ0pxq “

ÿ

kPZdppkq ϕ0pAx´Hkq, x P Rd

for some A,H P GLdpRq, with A diagonal, min1ďmďd

pApm,mqq ą 1, and some multi-vector

p withÿ

kPZdppkq “ detpAq,

28


we have that, for every multi-index α “ pα1, . . . , αdq P Ndzt0u,

ż

Rdxα ϕ0pxq dx “

1

detpAq pdiagpAqα ´ 1q

ÿ

0ďβăα

cαpβq

ż

Rdxβ ϕ0pxq dx

where

cαpβq :“

ˆ

α

β

˙

ÿ

kPZdppkq pHkqα´β, 0 ď β ă α,

xα “dź

m“1

xαmm , |α| “dÿ

m“1

αm,

and 0 ď β ă α meaning that, componentwise, all the components of β are not greaterthan the components of α, with at least one component strictly lesser. Moreover, Remark1.36 still holds. Proposition 1.37 is more tricky to extend to higher dimensions. Withd “ 2, if we consider a diagonal dilation matrix A P GL2pRq, with max

m“1,2Apm,mq ą 1,

an initial set of regular knots is of the form

t0 “ H Z2 with H “

„

h1 h2 cospθ1q

0 h2 cospθ1q

„

cospθ2q sinpθ2q

´ sinpθ2q cospθ2q

,

where θ1 is the angle between the two main axes, θ2 the angle describing the rotationof the system and h1, h2 ą 0 the sizes of the intervals between two consecutive knots oneach main axis. Indeed, we have that, for every j P N, the knots of tj “ A´jt0 belongto tj`1, and, as in Remark 1.3, for every k P Z2, t0 ´Hk “ t0p¨ ´ kq. In contrast withthe univariate case, with these knots we can have different quadrilateral and triangularmeshes. This creates different sequences of approximants to the limit function and ingeneral one should prove analogous of (1.33) for each of these choices. In the bivariatecase, however, is not hard to prove that the integral of a limit function f obtainedstarting from the compactly supported data f0 P `pZ2

q satisfies

ż

R2

fpxq dx “ h1 h2 sinpθ1qÿ

kPZ2

f0pkq,

independently from the considered mesh. 3

When dealing with semi-regular schemes, due to (1.14), we realize that we alreadyknow all the moments of most of the basic limit functions. Indeed, if we consider asemi-regular scheme over the initial mesh t0 in (1.3), h`, hr ą 0, with the subdivisionmatrix P, the left and right regular masks p`, pr, and k`pPq, krpPq in (1.7), then thebasic limit functions ϕk, k ď k`pPq are also basic limit functions of the regular schemedefined by the mask p` over the mesh h`Z, and similarly for ϕk, k ě krpPq and pr.Thus, we only need to compute the moments of ϕk, k`pPq ă k ă krpPq and to do so wecan exploit the knowledge of the moments of the other regular basic limit functions ofthe same subdivision scheme.

29


Proposition 1.40. For every α ě 0,

PT

„ż

Rxα ϕkpxq dx

kPZ“ 21`α

„ż

Rxα ϕkpxq dx

kPZ. (1.35)

In particular, for every α P N0,

Aα

„ż

Rxα ϕkpxq dx

k`pPqăkăkrpPq

“ bα, (1.36)

where

bα “

¨

˝

ÿ

kďk`pPq

`ÿ

kěkrpPq

˛

‚

ż

Rxα ϕkpxq dx Pirrpk, :q

T

and the matrixAα “ 21`α I ´ Pirrpk`pPq ` 1 : krpPq ´ 1, :qT

is invertible.

Proof. Since the columns of P are compactly supported, we can multiply the refinementequation (1.17) by xα and integrate componentwise, thus obtaining

„ż

Rxα ϕkpxq dx

kPZ“ PT

„ż

Rxα ϕkp2xq dx

kPZ, x P R.

Then the substitution y “ 2x on the right-hand side leads to (1.35). Moreover, due to(1.7), if we select on both sides of (1.35) the rows with indeces between k`pPq and krpPqwe obtain

21`α

„ż

Rxα ϕkpxq dx

k`pPqăkăkrpPq

“ PTirr

„ż

Rxα ϕkpxq dx

kPZ.

Isolating the contribution of the pα ` 1q-th moments of ϕk, k`pPq ă k ă krpPq, on theleft-hand side we get to the linear system in (1.36). Now, we observe that Pirrpk`pPq`1 :krpPq ´ 1, :q is the submatrix of P obtained by eliminating the first and the last rowsand columns of P. Since the first and last columns of P are two of its eigenvectors ofthe form

Pp:, 1q “

„

Pp1, 1q0

and

Pp:, krpPq ´ k`pPq ` 1q “

„

0

PpkrpPq ´ k`pPq ` 1, krpPq ´ k`pPq ` 1q

,

the spectrum of Pirrpk`pPq ` 1 : krpPq ´ 1, :q is contained in the spectrum of P and inparticular the dominant eigenvalue of Pirrpk`pPq ` 1 : krpPq ´ 1, :q does not exceed 1(Theorem 1.11 and Proposition 1.9). This is sufficient to guarantee the invertibility ofthe matrix Aα, for every α P N0.

30


Remark 1.41. Even if the right-spectrum of a semi-regular subdivision matrix P is dis-crete (Proposition 1.9), its left-spectrum contains at least the interval r2,8q. It alsocontains 1 due to Proposition 1.21. 3

1.2.2 Gramian and Cross-Gramian Matrices

Let us start considering a simple example of two regular convergent subdivision schemesover the same regular mesh t0 “ hZ, h ą 0, with the basic limit functions tϕkukPZ,tζkukPZ, compactly supported masks p, z, subdivision matrices P, Z and symbols ppωq,zpωq, respectively. The main tools for computing the cross-Gramian matrix

Gpk,mq “

ż

Rϕkpxq ζmpxq dx, k, m P Z, (1.37)

are given in Propositions 1.27 and 1.21.

Proposition 1.42. The cross-Gramian matrix G in (1.37) is a band-limited Toeplitzmatrix. Moreover, Gpk, 0q “ γ0phkq, k P Z, where

pγ0pωq “ pphω{2q zphω{2q pγ0pω{2q, ω P R. (1.38)

Proof. We start by observing that, for every k,m P Z,

Gpk,mq “

ż

Rϕkpxq ζmpxq dx

“

ż

Rϕ0px´ hkq ζ0px´ hmq dx

“

ż

Rϕ0py ´ hpk ´mqq ζ0pyq dy “ Gpk ´m, 0q.

Thus, G is a Toeplitz matrix. Now, consider η0 “ ϕ0p´¨q. It is easy to see that η0

satisfies the refinement equation on the Fourier side (1.26) with respect to the symbolppωq. Thus, by Proposition 1.27, we get

Gpk, 0q “

ż

Rϕ0px´ hkq ζ0pxq dx “

ż

Rη0phk ´ xq ζ0pxq dx “ γ0phkq, k P Z,

with γ0 satisfying (1.38).

Proposition 1.42 shifts the problem of computing the entries of G to the problem ofevaluating a certain basic limit function γ0 at the knots of the initial mesh t0 “ hZ,h ą 0, and we already know how to do it by Proposition 1.21.

Remark 1.43 ([51]). When ζ0 “ ϕ0, γ0 is the so-called autocorrelation function of ϕ0.In this case, we have that gpωq “ |ppωq|2, a real symmetric non-negative trigonomet-ric polynomial and so the mask g is symmetric. Moreover, γ0 “ }ϕ0}

22. If ϕk’s are

31


orthonomal, then

γ0p0q “ 1 and γphkq “

ż

R

ϕ0pxqϕ0px´ hkq dx “ 0, k P Zzt0u,

and so the scheme associated to γ0 is an interpolatory scheme. The converse is alsotrue. 3

Suppose now that the schemes at hand are semi-regular over the same semi-regularmesh t0 in (1.3) with h`, hr ą 0. We already know most of the entries of the correspond-ing cross-Gramian matrix G from the regular case, since, far from 0 we are still workingwith regular subdivision. The situation we have to deal with is the following:

G “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

. . .

. . .

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

+

npPq

looooomooooon

npZq

where the green (light) area represents the values that we already know from the regularcase and the blue (dark) area represents the unknown entries of G, where

npPq “ krpPq ´ k`pPq ´ 1 and npZq “ krpZq ´ k`pZq ´ 1,

are the numbers of the irregular basic limit functions of the two schemes respectively.The size of the blue area of course depends on the size of the supports of the irregularbasic limit functions of the two schemes. Since they also have compact support, seeProposition 1.9, the number of unknown entries of G is bounded by

p npPq ` npZq ` | supppp`q| ` | supppprq| ` | supppz`q| ` | supppzrq| ´ 11 q2 .

As we will see in a moment the number of unknowns is not relevant. It is crucial that

32


there are finitely many of them. The other ingredient of the method in Proposition 1.44is the following relation, already observed in [48]. Using the refinement equation (1.17)for both schemes we obtain

G “

ż

RrϕkpxqskPZ rζmpxqs

TmPZ dx

“

ż

RPT

rϕkp2xqskPZ rζmp2xqsTmPZ Z dx

“1

2PT G Z.

(1.39)

This relation yields a linear system of equations for the unknown entries of G. We provethat the knowledge of the regular entries of G determines the unknown entries uniquely.

Proposition 1.44. The finite system of linear equations obtained from (1.39) is uniquelysolvable for the irregular entries of G.

Proof. Seeking a contradiction, suppose there is another bi-infinite matrix F such thatF differs from G only on the irregular part and

F “1

2PT F Z.

We then consider the matrix ∆ “ G ´ F ı 0. By the linearity of the matrixmultiplication

∆ “ G´ F “1

2PT

pG´ Fq Z “1

2PT ∆ Z. (1.40)

Let n1 ă minpk`pPq, k`pZqq ă maxpkrpPq, krpZqq ă n2 such that rG “ Gpn1 : n2, n1 : n2q

contains all the irregular entries of G. If we consider analogously r∆, rP and rZ, we havethat r∆ contains all the non-zero elements of ∆ and, since, as in Proposition 1.9, rP andrZ contain all the non-zero elements of the corresponding rows of P and Z, respectively.We get an equivalent finite version of (1.40), namely,

r∆ “1

2rPT

r∆ rZ. (1.41)

Now, if r∆ ı 0, there exist k,m P N such that r∆pk,mq ‰ 0. Consider the vectors epkq

and epmq of the canonical basis. From (1.41), we have

0 ‰ r∆pk,mq “ pepkqqT r∆ epmq “1

2jpepkqqT prPj

qTr∆ rZj epmq .

Now, rP and rZ have P and Z, respectively, as submatrices on their diagonals, see Propo-sition 1.9 and Proposition 1.9. Thus, rP and rZ share the spectral properties of P and Z,

33


respectively. In particular rP and rZ both have dominant eigenvalue 1 with multiplicityone. So there exist 0 ă CpPq, CpZq ă 8 such that

$

&

%

}pepkqqT prPTqj} ď CpPq

}prZqj epmq} ď CpZq

, j P N.

This means that

0 ă |pepkqqT r∆ epmq| ď

ˇ

ˇ

ˇ

ˇ

1

2jpepkqqT prPj

qTr∆ prZqj epmq

ˇ

ˇ

ˇ

ˇ

ď1

2jCpPq CpZq }r∆} ÝÑ

jÑ`80,

which leads to a contradiction.

34

2 Wavelet Tight Frames

The concept of a frame was introduced by Duffin and Schaeffer in the fifties [28] asa generalization of the idea of a basis of an inner product vector space. In the lasttwenty years, frames, in particular wavelet frames, found a place in a wide spectrum ofapplications such as audio, image and surface compression, edge detection, inpainting,approximation of PDE solutions. Frames are flexible and efficient in implementations.The idea behind a wavelet frame is an efficient representation of the functions of an innerproduct vector space that splits the functions in a way convenient for their analysis andother manipulations.

Definition 2.1. Let V be an inner product vector space with the norm } ¨ } and F “

tψjujPI Ă V with the index set I at most countable. F is a frame for V if and only if

spanpFq}¨}“ V and there exists 0 ă A ď B ă 8 such that

A }f}2 ďÿ

jPI

| x f, ψj y |2ď B }f}2, f P V .

If A “ B the frame is said to be tight.

Remark 2.2. If a frame F is tight, then w.l.o.g. A “ B “ 1. Indeed, F{?A is still a

tight frame for which the Parseval’s equality holds

ÿ

jPI

ˇ

ˇ

ˇ

ˇ

B

f,ψj?A

Fˇ

ˇ

ˇ

ˇ

2

“ }f}2, f P V . (2.1)

Moreover, in this case we have the so called perfect reconstruction property, i.e.

f “ÿ

jPI

B

f,ψj?A

F

ψj?A, f P V , (2.2)

where the equality is intended in the norm } ¨ }. 3

Of course, orthonormal bases are frames, and in particular tight frames, but the notionof a frame allows for weaker assumptions on the elements of F which for example canbe linearly dependent, see Example 2.3.

Example 2.3. Let V “ R2 and consider the set

F “

"

f1 “

„

10

, f2 “

„

´1{2?

3{2

, f3 “

„

´1{2

´?

3{2

*

.

35


Then, for every x “ rx1, x2s P R2,

| x x, f1 y |2` | x x, f2 y |

2` | x x, f3 y |

2“

“ x21 `

ˆ

´x1

2`

?3

2x2

˙2

`

ˆ

´x1

2´

?3

2x2

˙2

“3

2x2

1 `3

2x2

2 “3

2}x}2.

Thus, F is a tight frame for R2. 4

The linear dependence of the frame elements introduces redundancy which, even ifnot always nice in theory, is very useful in applications. It adds robustness to noise whenthe information about a function is encoded via its frame coefficients, i.e. the valuestxf, ψjyujPI . Since in applications most of the functions/signals are compactly supportedand bounded, it is very convenient to see them as elements of L2

pRdq, d P N, which is

a Hilbert space and, in particular, an inner product vector space. We again focus onthe univariate case d “ 1. On one hand, the concept of a frame is very general. On theother hand, we would like to have some exploitable structure both for the constructionof frames and for their application. The most convenient way to this goal is via theso-called multi-resolution analysis. We use here the general definition introduced byChui, He and Stockler in [14, 15].

Definition 2.4. A family tVjujPN0 of closed subspaces of L2pRq is said to be a multi-

resolution analysis for L2pRq if the following conditions are satisfied.

(i) Vj´1 Ă Vj, for all j P N; (increasing subspaces)

(ii)ď

jPN0

Vj}¨}L2

“ L2pRq; (completessness in L2

pRq)

(iii) there exist families of functions Φ0 “ rφkskPZ and Φj “ rφj,kskPZ, j P N, such that

spanpΦjqL2

“ Vj, j P N0,

and, for every j P N, Φj´1 “ PTj´1Φj, for some matrix Pj´1;

(iv) there exists a vector c0 such that cT0 Φ0 ” 1.

The index j is called resolution level.

Definition 2.5. Consider a multi-resolution analysis in Definition 2.4. If there exists afamily of matrices tQjujPN such that the set F “ Φ0 Y tΨj “ QT

j ΦjujPN is a tight frame

for L2pRq, then F is called wavelet tight frame. The elements of Φ0 are called scaling

functions and the elements of Ψj framelets (wavelets if F is orthonormal). Furthermore,

36


F is said to be Cs, s ą 0, if all its elements belong to CspRq and to have v P N vanishingmoments if, for every j P N, k P Z,

ż

Rxα ψj,kpxq dx “ 0, α P t0, . . . , v ´ 1u. (2.3)

The smoothness and the number of vanishing moments of F will be crucial in Chapter3. Roughly speaking, if a function f P L2

pRq is regular, then locally it can be approxi-mated by a high degree polynomial. Thus, if the wavelet tight frame has a high numberof vanishing moments, the frame coefficients of f with respect to the framelets will benegligible. This fact means, on one hand, that one can approximate signals threshold-ing the frame coefficients and, on the other hand, that from the decay of the framecoefficients of f one can extract the information about the smoothness of f .

The simplest example of such a function system is the so called Haar system [36],which is orthonormal. The Haar system in its simplicity is not so desirable since it lacksboth smoothness and vanishing moments. For other interesting wavelet tight frames withnicer properties one had to wait until the seminal works of Daubechies, Meyer, Mallatand others at the end of the eighties (see e.g. [20, 49]) and what came afterwards.

Example 2.6 (Haar system). Consider a set of non-trivial intervals tIk “ rak, ak`1qukPZwith disjointed interiors and that form a partition of R. For every interval, we considerthe function φkpxq “ χIkpxq{

a

|Ik|. Of course,

ÿ

kPZ

a

|Ik| φkpxq ” 1, x P R.

We then consider the dyadic cuts of the intervals tIkukPZ namely tI1,kukPZ, closed on theleft and open on the right, such that

Ik “ I1,2k Y I1,2k`1, |I1,2k X I1,2k`1| “ 0 and |I1,2k| “ |I1,2k`1| “|Ik|

2,

and the corresponding indicator functions φ1,k “ χI1,kpxq{b

|I1,k|. Iterating this process,

we obtain tΦjujPN0 such that

Φj´1 “1?

2

»

—

—

—

—

—

—

—

–

. . . 11

11

1. . .

1

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

T

Φj, j P N.

Moreover, the closed subspaces tVj “ spanpΦq}¨}L2

ujPN0 form a multi-resolution analysis

for L2pRq. Indeed for every k P Z,

ď

jPN0

Vj contains all the simple functions defined on

37


all dyadic subintervals of Ik which are dense in L2pIkq. Let

Ψj “1?

2

»

—

—

—

—

—

—

—

–

. . . 1´1

1´1

1. . .

´1

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

T

Φj, j P N.

It is easy to see that F “ Φ0Y tΨjujPN is an orthonormal set. Moreover, since for everyj P N,

Vj “ spanpΦj´1 YΨjq}¨}L2

“ spanpΦ0 Y tΨmujm“1q

}¨}L2

ÝÑjÑ8

L2pRq,

the orthonormality of F , implies that F is a wavelet tight frame for L2pRq. In partic-

ular, the elements of F are not continuous and F has only one vanishing moment. Inparticular, we observe that, when Ik “ rk, k ` 1q, k P Z, we obtain a shift-invariantsystem with φk “ φ0p¨ ´ kq, where

pφ0pωq “1` e´2πiω

2pφ0pω{2q, ω P R.

4

Whenever, for every j P N0, Pj ” P0, Qj ” Q1 and Φj “ 2j{2Φ0p2j¨q, in Definition

2.4 and 2.5, we have that

Φ0 “ 2j{2 pPj0qT Φ0p2

j¨q and Ψj “ 2j{2Q1 Φ0p2

j¨q “ 2pj´1q{2Ψ1p2

j´1¨q. (2.4)

At this point the first of these two equations should remind us of the refinement equation(1.17). Indeed, a good candidate for the set of scaling functions is the set of basic limitfunctions of a subdivision scheme, after a proper renormalization.

Remark 2.7. The role of the framelets Ψj is to describe the features needed to pass fromresolution level j ´ 1 to resolution level j. Indeed, for f P L2

pRq, due to (2.2) we havethat the sequence

f0 “ x f, Φ0 yT Φ0 P V0

fj “ fj´1 ` x f, Ψj yT Ψj P Vj, j P N,

converges to f in }¨}L2 . Moreover, due to Definition 2.4 (iii) and (iv), it is always possibleto ensure that the framelets have at least one vanishing moment. Thus, Definition 2.5is consistent. 3

In the regular case, where the underlying structure is shift-invariant, we can use

38


powerful tools from the literature, namely the Unitary Extension Principle [55, 56] andthe Oblique Extension Principle [16, 24], to construct wavelet tight frames, see Section2.1. These constructions, in our case, are based on the symbols of convergent subdivisionshcemes. In the semi-regular case, due to the lack of shift invariance, those tools are notavailable in the same form. Indeed, we leave the Fourier domain and use the generalframework provided in [14, 15], where the construction method works directly on thematrices involved. After reviewing these methods, we present a first example of semi-regular wavelet tight frame with two vanishing moments based on the cubic B-splinescheme, see Section 2.1.1. This example illustrates the difficulties of the OEP typeconstruction. In Section 2.2, we propose a family of semi-regular wavelet tight framesbased on the Dubuc-Deslauriers interpolatory schemes whose construction overcomesthose difficulties and is UEP-based.

2.1 The Unitary and Oblique Extension Principles: fromSymbols to Matrices

We start with the regular case. Consider a convergent regular subdivision scheme overthe initial mesh t0 “ hZ, h ą 0, with the basic limit functions rϕk “ ϕ0p¨ ´ hkqskPZ,the compactly supported mask p, the symbol ppωq and the subdivision matrix P. Totransform our basic limit functions into the scaling functions we need to renormalizethem in the following way. We define

Φ0 “ rφkskPZ “ D´1{2rϕkskPZ, Dpk,mq “

$

&

%

ż

Rϕkpxq dx, k “ m,

0, otherwise.(2.5)

Moreover, from (2.4) using (1.17) we get

21{2 PT0 Φ0p2¨q “ Φ0 “ D´1{2

rϕkskPZ “ D´1{2 PTrϕkp2¨qskPZ

“ D´1{2 PT D1{2rϕkp2¨qskPZ “ D´1{2 PT D1{2 Φ0p2¨q.

(2.6)

Thus, P0 “ 2´1{2D1{2PD´1{2. In particular, in the regular case, from Section 1.2.1, weknow that Dpk, kq ” h, so P0 “ 2´1{2P. Next we would like is to find a matrix Q1 suchthat

F “ Φ0 Y

Ψj “ 2j{2 QT1 Φ0p2

j¨q(

jPN (2.7)

is a wavelet tight frame for L2pRq. It would be also preferable that the shift-invariant

structure is kept also with respect to the framelets. In particular, we would like that

ψ1,kpxq “ ψ1,m

ˆ

x´ hk ´m

M

˙

, for k ” m mod M,

39


for some M P N. This is reflected on the Q1 in the following way:

Q1pk, n`m´ 1q “ 2´1{2 qmpk ´ 2jq, k, n P Z, m “ 1, . . . ,M

for some vectors qm, m “ 1, . . . ,M . Asking the ψj,k to be compactly supported thenmeans that the vectors tqmu

Mm“1 must be compactly supported. The following result

characterizes all the wavelet tight frames that have this form.

Theorem 2.8 (Oblique Extension Principle (OEP), [16, 24]). The set F in (2.7) is awavelet tight frame with v P N vanishing moments if and only if there exists

spωq “snpωq

sdpωq, ω P R,

with snpωq, sdpωq trigonometric polynomials, continuous at 0 with sp0q “ 1 such that

$

’

’

’

’

’

’

&

’

’

’

’

’

’

%

sp2ωq ppωqppωq `Mÿ

m“1

qmpωqqmpωq “ spωq,

sp2ωq ppωqppω ´ 1{2q `Mÿ

m“1

qmpωqqmpω ´ 1{2q “ 0,

a.e., (2.8)

for

qmpωq “1

2

ÿ

kPZ

qmpkq e´2πikω, m “ 1, . . . ,M,

and

qmpωq “

ˆ

1´ e´2πiω

2

˙v

qrvsm pωq, m “ 1, . . . ,M,

for some trigonometric polynomials tqrvsm pωqum“1,...,M ,

The Unitary Extension Principle (UEP) [55, 56] is a particular case of the OEP whenspωq ” 1. In general, this restriction yields wavelet tight frames with at most onevanishing moment, except for special cases. On the other hand, the UEP is much easierto handle. For instance, this simpe and interesting result holds.

Theorem 2.9. Let ppωq and zpωq be the symbols of convergent regular subdivisionschemes satisfying (2.8) with spωq ” 1, trigonometric polynomials tqmpωqum“1,...,Mp,tbmpωqum“1,...,Mz , and vp, vz P N, respectively. Then the symbol ppωqzpωq satisfies (2.8)with s ” 1 together with the trigonometric polynomials

t ppωq bmpωq um“1,...,Mz Y t zpωq qmpωq um“1,...,Mp Y t qm1pωq bm2pωq um1“1,...,Mp,m2“1,...,Mz

and v “ mintva, vpu.

Proof. The proof simply follows by multiplying the systems arising from (2.8) for ppωqand for zpωq, term by term.

40


Remark 2.10. The strategy of Theorem 2.9 in general is not valid for the OEP case,when spωq ı 1. A counterexample is presented in Remark 2.22. 3

Theorem 2.9 can be used for example to obtain explicit algebraic expressions forframelets with 1 vanishing moment for the family of B-spline schemes, since the B-splinesymbols are powers of p1` e´2πiω

q{2, which is the symbol associated to the regular Haarsystem, see Example 2.6. This is an alternative, more straightforward way for obtainingthe framelets in [8] in the regular case.

Example 2.11 (Cubic B-spline, part III). As we saw in Example 1.31, the symbolassociated to the cubic B-spline scheme is

ppωq “

ˆ

1` e´2πiω

2

˙4

, ω P R,

up to a unitary factor, see Remark 1.24. In particular, ppωq is the fourth power of theHaar symbol. It is easy to see that the Haar symbol also satisfies (2.8) with spωq ” 1,v “ 1 and one trigonometric polynomial q1pωq “ p1 ´ e´2πiω

q{2. We can use Theorem2.9 three times ending up essentially with the following four trigonometric polynomials,

qm “

d

ˆ

4

m

˙ ˆ

1´ e´2πiω

2

˙m ˆ

1` e´2πiω

2

˙4´m

, m “ 1, . . . , 4.

In the end, the corresponding vectors are

q1 “

»

—

—

—

—

–

´1{4´1{2

01{21{4

fi

ffi

ffi

ffi

ffi

fl

, q2 “?

6

»

—

—

—

—

–

1{80

´1{40

1{8

fi

ffi

ffi

ffi

ffi

fl

, q3 “

»

—

—

—

—

–

´1{41{2

0´1{2

1{4

fi

ffi

ffi

ffi

ffi

fl

, q4 “

»

—

—

—

—

–

1{8´1{2

3{4´1{2

1{8

fi

ffi

ffi

ffi

ffi

fl

.

4

Before diving into the semi-regular case, let us take a look at the UEP conditions froma different angle. First of all, we can interpret a symbol ppωq as the scalar product oftwo particular vectors

ppωq “1

2

ÿ

kPZ

ppωq e´2πikω“

1

2re´2πikω

skPZ p. (2.9)

41


The same holds for all the tqmpωqum“1,...,M . Thus, we write

$

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

%

2 “ ppωqppωq `Mÿ

m“1

qmpωqqmpωq

“1

2re´2πikω

sTkPZ

´

ppT `“

q1, . . . ,qM‰ “

q1, . . . ,qM‰T

¯

re2πikωskPZ,

0 “ ppωqppω ´ 1{2q `Mÿ

m“1

qmpωqqmpω ´ 1{2q

“1

2re´2πikω

sTkPZ

´

ppT `“

q1, . . . ,qM‰ “

q1, . . . ,qM‰T

¯

rp´1qke2πikωskPZ,

a.e.,

where we multiplied by 2 both equations in (2.8) and set spωq ” 1. Now we observe thatppT {2 is one of the elements of the rank-1 expansion of P0P

T0 , and the same holds for

rq1, . . . ,qM srq1, . . . ,qM sT{2 and Q1Q

T1 . In particular, if

B0 :“ ppT `“

q1, . . . ,qM‰ “

q1, . . . ,qM‰T, (2.10)

we have thatÿ

kPZ

B0p¨ ´ 2k, ¨ ´ 2kq “ P0 PT0 ` Q1 QT

1 . (2.11)

Moreover, from the first UEP condition we have that the sum of each of the diagonalsof B0 gives one of the coefficients of the polynomial 2, i.e.

ÿ

kPZ

B0pk, kq “ 2 andÿ

kPZ

B0pk, k `mq “ 0, m P Zzt0u. (2.12)

The second UEP condition implies that on each diagonal of B0 the sum of the elementsat odd positions is the same as the sum of the elements at even positions, i.e.,

ÿ

kPZ

B0p2k, 2kq “ÿ

kPZ

B0p2k ` 1, 2k ` 1q,

ÿ

kPZ

B0p2k, 2k `mq “ÿ

kPZ

B0p2k ` 1, 2k ` 1`mq, m P Zzt0u.(2.13)

Due to (2.12), the first identity in (2.13) must be equal to 1, while the second identity(2.13) must be always 0. Thus, (2.10) becomes

I “ P0 PT0 ` Q1 QT

1 . (2.14)

Thus, in the regular case, finding solutions to the UEP conditions is equivalent to finda block 2-slanted symmetric factorization of the matrix I´P0P

T0 .

42


Remark 2.12. Due to (2.14), a necessary condition for the costruction of a wavelet tightframe in the regular case is that I´P0P

T0 is a positive semi-definite matrix. 3

We can go deeper from (2.14). If we multiply both sides in (2.14) by ΦjpxqT from the

left and by Φjpyq from the right and use (2.4), we obatin

ΦjpxqT Φjpyq “ Φjpxq

T P0 PT0 Φjpyq ` Φjpxq

T Q1 QT1 Φjpyq

“ Φj´1pxqT Φj´1pyq ` Ψjpxq

T Ψjpyq, x, y P R.(2.15)

The quantity ΦjpxqT Φjpyq is a kernel which defines a projection from L2

pRq onto theelement of the associated multi-resolution analysis Vj, i.e

Kj : L2pRq ÝÑ Vjf ÞÝÑ

ż

Rfpxq Φjpxq

T Φjp¨q dx.(2.16)

This point of view does not depend on symbols or masks, but only on the family of func-tions and matrix refinement relation between them. Indeed, under suitable assumptions,this approach can be extended to include the semi-regular case and even the irregularone. The more general OEP conditions can also be incorporated. This is the essence ofthe works [14, 15] that we are going to briefly review, focusing on the key points neededfor our further construction.

In the general case, we consider a family of vectors of refinable functions tΦjujPN0

that defines a multi-resolution analysis in Definition 2.4. To proceed we need this familyto satisfy the following assumptions. Assumption 1 requires uniform behaviour for thefunctions defining the multi-resolution analysis, even if the setting is not shift-invariant.

Assumption 1. For every j P N0,

(a) Φj is a Riesz basis for Vj, i.e. there exist 0 ă Aj ď Bj ă 8 such that

Aj }f}2`2 ď }ΦT

j f }2L2 ď Bj }f}2`2 , @f P `2

pZq.

(b) Φj is uniformly bounded, i.e. supkPZ}φj,k}L8 ă 8.

(c) Φj is strictly local, i.e. supkPZ| supppφj,kq| ă 8 and there exists mj ă 8 such that,

for every index set I Ă Z with |I| ą mj,

č

kPIsupppφj,kq “ H.

(d) limjÑ8

supkPZ

| supppφj,kq| “ 0.

43


Remark 2.13. Conditions (b) and (c) of Assumption 1, imply that, for every j P N0, thefamily Φj is a Bessel sequence, i.e., there exists a constant CB,j ą 0 such that

ÿ

kPZ

|x f, φj,k y|2ď CB,j }f}

2L2 , f P L2

pRq.

Indeed, for every f P L2pRq,

ÿ

kPZ

|x f, φj,k y|2ď psup

kPZ}φj,k}L8q

2ÿ

kPZ

ż

supppφj,kq

|fpxq|2 dx,

where supkPZ}φj,k}L8 is bounded due to (b) and the sum is bounded by Cpmjq}f}

2L2 , for

some Cpmjq ą 0, due to (c). 3

The second assumption is a condition on the Gramian matrices

Gj :“

ż

RΦjpxq Φjpxq

T dx, j P N0.

Assumption 2. There exists v P N such that, for every j P N0,

ż

Rxα Φjpxq G´1

j Φjpyq “ yα, y P R, α P t0, . . . , v ´ 1u.

Remark 2.14. Assumption 2 is linked to the degree of the polynomial space that thefunctions Φj are able to span. Indeed, a necessary condition for Assumption 2 is thatthere exist vectors tcj,αu

vα“0 such that

cTj,α Φjpxq “ xα, x P R.

In particular, if we denote by mj,α the vector of the pα ` 1q-th moment of Φj, then

G´1j mj,α “ cj,α. (2.17)

3

The last assumption is related, roughly speaking, to the primitives of the functions Φj

and, together with Assumption 2, is the one closely related to the vanishing moments ofthe resulting wavelet tight frame.

Assumption 3. There exists w P N such that there exists a family of scaling func-tions tΦ

r´wsj ujPN0 that generates a multi-resolution analysis in Definition 2.4 where Φ

r´wsj

satisfies Assumption 1 and the following conditions, for every j P N0.

(a) for every k P Z, Φr´wsj Ă Hw

pRq, where Hw denotes the Sobolev spaces of squareintegrable functions with w square integrable weak derivatives.

44


(b) f P Vj has w vanishing moments if and only if there exists f P `2pZq such that

fpxq “ fTdw

dxwΦr´wsj pxq , x P R.

Moreover, f decays exponentially if f does.

As we will see in the following constructions, Assumptions 1-3 are easily satisfied inreal applications.We are now able to state the fundamental result of Chui, He, Stockler which characterizeswavelet tight frames in a generic setting.

Theorem 2.15 ([15]). Under Assumption 1, a set of bi-infinite matrices tQjujPN definesa wavelet tight frame in Definition 2.5 if and only if there exists a set of bi-infinitesymmetric semi-positive definite matrices tSjujPN such that:

(i) for every j P N0, there exists Cj ą 0 such that

ż

R2

fpxq ΦjpxqT Sj Φjpyq fpyq dx dy ď Cj }f}

2L2 , f P L2

pRq;

(ii) for every f P L2pRq,ż

R2

fpxq ΦjpxqT Sj Φjpyq fpyq dx dy ÝÑ

jÑ8}f}2L2 ;

(iii) for every j P N,Sj ´ Pj´1 Sj´1 PT

j´1 “ Qj QTj .

Moreover, if Assumptions 2 and 3 are satisfied with v “ w P N, then the correspondingwavelet tight frame has v vanishing moments if and only if, for every j P N0, the matrixSj satisfies

(a) }Sj}`2Ñ`2 ă 8;

(b) DC ą 0, r ą 2v ` 1:

ˇ

ˇ ΦjpxqT Sj Φjpyq

ˇ

ˇ ďC

p1` |x´ y|qr, x, y P R;

(c)

ż

Rxα Φjpxq

T`

G´1j ´ Sj

˘

Φjpyq dx “ 0, a.e., α P t0, . . . , v ´ 1u;

(d)

ż

Ryα

ż x

´8

px´ tqv´1

pv ´ 1q!Φjptq

T`

G´1j ´ Sj

˘

Φjpyq dt dy “ 0 a.e., α P t0, . . . , v ´ 1u.

Remark 2.16. If one desires a wavelet tight frame with compactly supported elements,due to Theorem 2.15 (iii), the matrices tQjujPN must have compactly supported columns,which means Sj must be bandlimited, for every j P N. 3

45


Remark 2.17. Condition (d) in Theorem 2.15 is implied by condition (c) if one canexchange the order of integration. This, however, is not always the case, see e.g. Example2.11. On the other hand, for our construction in the semi-regular, which is based on alocal modification of Sj, (d) will be implied by (c) trivially. 3

Remark 2.18. In the regular case, when spωq in (2.8) is a symmetric polynomial, theresulting matrices Sj are equal to the Toeplitz matrix obtained from the coefficients ofspωq. 3

Theorem 2.15 basically gives us the roadmap for the construction of a wavelet tightframe:

1st: choose a suitable multi-resolution analysis satisfying Assumptions 1, 2 and 3;

2nd: find a suitable set of matrices tSjujPZ satisfying Theorem 2.15;

3rd: take the square root of the matrices Sj ´Pj´1Sj´1PTj´1.

We already have good candidates for the 1st step - the renormalized basic limit functionsof subdivision schemes. We still need to check the properties required by Assumptions1, 2 and 3, which we will do case by case when needed. For the second step, Theorem2.15 itself already gives us a clue on how to choose the matrices Sj. Indeed, if we areable to choose Sj such that

Sj mj,α “ cj,α, α P t0, . . . , v ´ 1u,

then Remark 2.14 guarantees that hypothesis (c) of Theorem 2.15 holds. Moreover,working in the semi-regular setting is advantageous since Sj ” S0, j P N, and additionallywe restrict the search to bandlimited matrices. In general, it is quite hard to factorize asemi-positive definite bi-infinite matrix, even if it is bandlimited and consists of positivesemi-definite blocks (2.11). The factorization requires the solution of several quadraticsystems. Thus, it is better to exploit the properties of the considered system case bycase.

With a general strategy at hand, we focus on the regular and semi-regular cases.In both cases we start from a convergent subdivision scheme with continuous basiclimit functions rϕkskPZ, which satisfy the refinement equation (1.17) with respect to asubdivision matrix P. First of all we still need to justify the renormalization (2.5). Inthe regular UEP case, we saw that it is always possible and it works just fine, translatingthe condition spωq ” 1 in (2.8) on the polynomial side to the condition S0 “ I in (2.11)on the matrix side. No doubt then that this is the natural renormalization one wants fordefining the scaling function. However, a question arises: is this renormalization alwayspossible? The answer to this question is yes if and only if we can prove that for theconsidered semi-regular convergent scheme we have

ż

Rϕkpxq dx ą 0, k P Z. (2.18)

46


This unfortunately is not the case (a counterexample can be found in Section 2.2.2,n “ 2). On the other hand, cases where the condition (2.18) fails are of no interestfor us because of Assumption 3. Indeed, if we are able to construct a wavelet tightframe with one vanishing moment, since the frame elements are linear combination ofthe scaling functions, Assumption 3 implies that there exists a convergent subdivisionscheme with basic limit functions whose derivatives are linear combinations of the basiclimit functions of the scheme considered to construct the scaling functions. As observedin [22], this is true if and only if there exists a monotone sequence b P `pZq such that

bT ∆ “1

2bT ∆ P and bpk ` 1q ´ bpkq “ C

ż

Rϕkpxq dx, k P Z, (2.19)

for some constant C ‰ 0, where

∆pk,mq “

$

&

%

p´1qk`m`1, if k ´ 1 ď m ď k,

0, otherwise.

This requires that all the integrals of ϕk must be w.l.o.g. strictly positive. Thus, if awavelet tight frame can be constructed via Theorem 2.15 starting from a convergentsubdivision scheme, the integrals of its basic limit functions must be positive in orderto perform the renormalization (2.5). Due to Proposition 1.40, we are able to computethose integrals.

Assumption 1, conditions (b), (c) and (d) are rather easy to check both in the regularand the semi-regular settings. The Riesz basis condition (a) is trickier to prove in general,but for the regular setting we have the following sufficient condition ensuring (b).

Proposition 2.19. Let G0 be the Gramian matrix of Φ0 “ rφ0,k “ φ0,0p¨ ´ hkqskPZ,h ą 0. If there exist 0 ă A0 ď B0 ă 8 such that

A0 ď gpωq “ÿ

kPZ

G0p0, kq e´2πihkω

ď B0, ω P R, (2.20)

then Φ0 is a Riesz basis for V0 with bounds A0 and B0.

Proof. Let f P `2pZq and consider f “ ΦT

0 f . Since the Fourier transform is a linearisometry on L2

pRq, we have

}f}2L2 “

ż

R| pfpωq|2 dω “

ż

R

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

kPZ

fpkq pφ0,kpωq

ˇ

ˇ

ˇ

ˇ

ˇ

2

dω

“

ż

R

ˇ

ˇ

ˇ

ˇ

ˇ

pφ0,0pωqÿ

kPZ

fpkq e´2πhkω

ˇ

ˇ

ˇ

ˇ

ˇ

2

dω.

47


Since the function F pωq “ÿ

kPZ

fpkq e´2πhkω is periodic with period 1{h, we can split R

intoď

kPZ

rk, k ` 1q{h and obtain

}f}2L2 “ÿ

kPZ

ż pk`1q{h

k{h

|pφ0,0pωq|2|F pωq|2 dω

“

ż 1{h

0

|F pωq|2ÿ

kPZ

ˇ

ˇ

ˇ

ˇ

pφ0,0

ˆ

ω ´k

h

˙ˇ

ˇ

ˇ

ˇ

2

dω.

(2.21)

Now, focusing on the internal sum, we observe that

ÿ

kPZ

ˇ

ˇ

ˇ

ˇ

pφ0,0

ˆ

ω ´k

h

˙ˇ

ˇ

ˇ

ˇ

2

“ÿ

kPZ

pφ0,0

ˆ

ω ´k

h

˙

pφ0,0

ˆ

ω ´k

h

˙

“ÿ

kPZ

ż

Rφ0,0pxq e

´2πixpω´ khq dx

ż

Rφ0,0pyq e

2πiypω´ khq dy

We can then rearrange the order of integration and substitute x “ z ` y to get

ÿ

kPZ

ˇ

ˇ

ˇ

ˇ

pφ0,0

ˆ

ω ´k

h

˙ˇ

ˇ

ˇ

ˇ

2

“ÿ

kPZ

ż

Re´2πizpω´ k

hqż

Rφ0,0pz ` yq φ0,0pyq dy dz

“ÿ

kPZ

pγ0

ˆ

ω ´k

h

˙

,

where

γ0pxq “

ż

Rφ0,0px` yq φ0,0pyq dy, x P R.

Finally, due to the Poisson summation formula (see e.g. [64]),

ÿ

kPZ

ˇ

ˇ

ˇ

ˇ

pφ0,0

ˆ

ω ´k

h

˙ˇ

ˇ

ˇ

ˇ

2

“ hÿ

kPZ

γ0 phkq e2πihkω

“ hÿ

kPZ

G0p0, kq e´2πihkω

“ h gpωq.

Thus, from (2.21), by hypothesis,

hA0

ż 1{h

0

|F pωq|2 dω ď }f}2L2 ď hB0

ż 1{h

0

|F pωq|2 dω.

48


To conclude the proof, we observe that

ż 1{h

0

|F pωq|2 dω “ÿ

k,mPZ

fpkq fpmq

ż 1{h

0

e´2πihωpk´mq dω

“1

h}f}2`2 .

Remark 2.20. Proposition 2.19 is equivalent to a property of the so-called bracket productrpφ0,0, pφ0,0s, see [39] and references therein. 3

In the regular case then, we only need to compute the Gramian matrix of Φ0 using theresults in Section 1.2.2 and then compute the bounds for the trigonometric polynomialin (2.20). For the other levels j P N, Φj “ 2j{2Φ0p2

j¨q and we get for free the Riesz

property, since

}ΦTj f}2L2 “

ż

R

ÿ

kPZ

|φj,kpxq fpkq|2 dx

“ 2jż

R

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

kPZ

φ0,kp2jxq fpkq

ˇ

ˇ

ˇ

ˇ

ˇ

2

dx “ }ΦT0 f}2L2 .

(2.22)

In the semi-regular setting, we use Proposition 2.19 to check that the regular schemes onthe left and on the right of t0p0q satisfy Assumption 1. Additionaly we need to guaranteethat the overall resulting semi-regular scheme satisfies Assumption 1. In particular, thelinear independence of the basic limit functions is enough to guarantee the Riesz basisproperty for the whole family of scaling functions.

Proposition 2.21. Let Φ0 be the vector of the scaling functions of a semi-regular multi-resolution analysis with the corresponding left and right regular schemes which inducemulti-resolution analysis satisfying Assumption 1. If Φ0 is linearly independent, then itis a Riesz basis for V0.

Proof. Let B`, Br ą 0 be the Riesz upper bounds for the left and right regular multi-resolution analysis, respectively. Then, for every f P `2

pZq,

}ΦT0 f}2L2 “

›

›

›

›

›

›

¨

˝

ÿ

kďklpPq

`ÿ

k`pPqăkăkrpP

`ÿ

kěkrpPq

˛

‚ fpkqφ0,k

›

›

›

›

›

›

2

L2

ď

ˆ

B` ` maxk`pPqăkăkrpPq

}φ0,k}2L2 ` Br

˙

}f}2`2 .

The lower estimate follows directly from the linear independence of Φ0.

49


We proceed further with the cubic B-spline scheme in Examples 1.31, 1.32 and 2.11.Using Theorem 2.15, we are searching for a wavelet tight frame with two vanishingmoments.

2.1.1 A First Example from Cubic B-spline

Regular Case

We consider the initial regular mesh t0 “ hZ, h ą 0. In Example 1.31, the regular cubicB-spline scheme produces piecewise cubic basic limit functions ϕk “ ϕ0p¨´hkq P C3´ε

pRq,ε ą 0, that satisfy a refinement equation (1.17) with respect to the subdivision matrixin (1.29). By Proposition 1.37, the integrals of ϕk are equal to h for all k P Z. Thus,

the renormalized scaling functions are φk “ ϕk{?h, k P Z. To compute the Gramian

matrix of the scaling functions in Φ0, we compute the Gramian matrix of the basic limitfunctions via Proposition 1.42 and then rescale. Indeed,

G0 “

ż

RΦ0pxq Φ0pxq

T dx “1

h

ż

RrϕkpxqskPZ rϕkpxqs

TkPZ dx “

1

hG.

The matrix G is a banded Toeplitz matrix and it is defined by the vector g obtained asa solution of

»

—

—

—

—

—

—

—

—

–

1{16 7{16 7{16 1{161{128 7{32 35{64 7{32 1{128

1{16 7{16 7{16 1{161{128 7{32 35{64 7{32 1{128

1{16 7{16 7{16 1{161{128 7{32 35{64 7{32 1{128

1{16 7{16 7{16 1{16

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

T

g “ g, (2.23)

with the sum of the elements of g equal to h. The entries of the matrix in (2.23) areobtained from the coefficients of the square of the symbol in Example 1.31. The solutionwe seek has sum of the elements equal to h, which leads to the unique vector

g “ h“

1{5040 1{42 397{1680 151{315 397{1680 1{42 1{5040‰T.

Thus, to prove the Riesz basis property, due to Proposition 2.19, we have to find boundsfor the function

gpωq “151

315`

397

840cosp2πωq `

1

21cosp4πωq `

1

2520cosp6πωq, ω P R.

The function g is periodic of period 1 and it satisfies

17

315“ g

ˆ

1

2

˙

ď gpωq ď gp0q “ 1.

50


Thus, the scaling functions Φ0 form a Riesz basis for the closure of their span V0. Then,by (2.22), our system Φ0 satisfies Assumption 1 (a). Conditions (b), (c) and (d) ofAssumption 1 are trivially satisfied.

The partition of unity property (1.16) implies that, for every j P N,

ΦjpxqT

?h

2j{21 “ Φ0p2

jxqT?h 1 “ rϕkp2

jxqsTkPZ 1 “ 1, x P R,

and we have

cj,0 “

?h

2j{21 and mj,0 “

ż

RΦjpxq dx “ 2j{2

ż

RΦ0p2

jxq dx “

?h

2j{21

satisfying (2.17). This is coherent with the tight frame construction (with one vanishingmoment) in Example 1.32, where Sj ” I. To get one more vanishing moment, however,we need to check Assumptions 2 and 3 for v “ w “ 2 and then choose a suitable S0 toapply Theorem 2.15.

Assumption 3 follows directly from (2.19) and it is a well known property of B-splines(see e.g. [26]). To check Assumption 2 with v “ 2, we observe that, from Example1.31, the right-eigenspace of P related to the eigenvalue 1{2 can be extended uniquely,as in Proposition 1.9, to the corresponding right-eigenspace of P with respect to thesame eigenvalue. The resulting right-eigenspace of P is formed by the samples of all thepolynomials of the form λx, λ P Rzt0u. In particular, rhkskPZ is a right-eigenvector of Pand we have, for every j P N0,

ΦjpxqT

?h

232jrkskPZ “ Φ0p2

jxqT?h

2jrkskPZ “ rϕkp2

jxqsTkPZ1

2jrkskPZ “ x, x P R.

Thus, Assumption 2 is satisfied with v “ 2 and, moreover, we get

cj,1 “

?h

232jrkskPZ and mj,1 “ Gj cj,1, j P N0.

In particular, we get mj,1 “ cj,1, j P N0. Unfortunately, even if the choice of S0 “ I fitsinto hypothesis (a), (b) and (c) of Theorem 2.15, it fails to meet (d) for α “ 1. Roughlyspeaking S0 is not a good enough approximation of G´1

0 . Indeed, as observed in [13] inthe regular case, to get v vanishing moments from the OEP (2.8) one has to choose ssuch that

spωq ´ gpωq´1“ Opω2v

q, ω Ñ 0,

with gpωq as in (2.20). One way to do that is to choose spωq to be the Taylor polynomialof degree 2v ´ 1 of g´1

pωq at ω “ 0. Then we can easily construct the matrices Sj inRemark 2.18. Since 0 ă gpωq ď 1 is real, we can exploit the Taylor expansion of the

51


function 1{p1´ xq at 0 to get

gpωq´1“

1

1´ p1´ gpωqq“

ÿ

jPN0

p1´ gpωqqj, ω Ñ 0

where the factor p1´ gpωqqj adds a polynomial term of lowest degree j, since gp0q “ 1.Thus, we can easily choose spωq such that

2v´1ÿ

j“0

p1´ gpωqqj “ spωq ` Opω2vq, ω Ñ 0.

In our case,

3ÿ

j“0

p1´ gpωqqj “ 1 `p2πωq2

3` Opω4

q

“5

3´

2

3

ˆ

1´p2πωq2

2

˙

` Opω4q

“5

3´

2

3cosp2πωq ` Opω4

q

“ ´1

3e´2πiω

`5

3´

1

3e2πiω

` Opω4q, ω Ñ 0,

so we can choose

Sj “

»

—

—

—

—

—

—

–

. . . ´1{35{3 ´1{3

´1{3 5/3 ´1{3

´1{3 5{3

´1{3. . .

fi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Remark 2.22. If we apply the same procedure for the linear B-spline scheme, see Exam-ples 1.5, 1.12, 1.20 and 1.25, to get two vanishing moments we obtain

s1pωq “ ´1

6e´2πiω

`4

3´

1

6e2πiω, ω P R.

If Theorem 2.9 could be applied to the OEP, then we should have

s1pωq2“ spωq ` Opω4

q, ω Ñ 0,

52


but

s1pωq2´ spωq “ ´

1

36e´4πiω

´1

9e´2πiω

`1

6´

1

9e2πiω

´1

36e´4πiω

“ ´1

9´

2

9p2πωq2 ` Opω4

q, ω Ñ 0.

3

Hence, we proceed differently. Having Pj “ P{?

2 as in (2.6), with P in Example1.31, to get the matrices Qj, we need to factorize

Rj “ Sj ´ Pj´1 Sj´1 PTj´1 “ S0 ´

1

2P S0 P.

The resulting matrix is the following

Rj “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

1{3841{96

7{384 1{96 1{3841{48 1{24 1{96

´7{128 1{48 7{384 1{96 1{384. . . ´59{96 ´1{8 1{48 1{24 1{96

79{64 ´59{96 ´7{128 1{48 7{384 1{96 1{384´59{96 4{3 ´59{96 ´1{8 1{48 1{24 1{96´7{128 ´59{96 79{64 ´59{96 ´7{128 1{48 7{384

1{48 ´1{8 ´59{96 4/3 ´59{96 ´1{8 1{48

7{384 1{48 ´7{128 ´59{96 79{64 ´59{96 ´7{1281{96 1{24 1{48 ´1{8 ´59{96 4{3 ´59{96

1{384 1{96 7{384 1{48 ´7{128 ´59{96 79{64

1{96 1{24 1{48 ´1{8 ´59{96. . .

1{384 1{96 7{384 1{48 ´7{1281{96 1{24 1{48

1{384 1{96 7{3841{96

1{384

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

To get the frame elements, we factorize Rj “ QjQTj . Since we want to keep the shift-

invariant structure and to have compactly supported framelets, we are searching firstfor a blocking decomposition of Rj as in (2.11), i.e. we want to find a finite positivesemi-definite block B such that

Rj “ÿ

kPZ

Bp¨ ´ 2k, ¨ ´ 2kq,

and then factorize B to obtain the vectors qm. From the structure of Rj we deducethat the smallest B possible must belong to R7ˆ7. Since the Fejer-Riesz Theorem states

53


that the OEP conditions (2.8) can always be satisfied with two polynomials q1, q2, thenthere exists a rank-2 B that does what we seek. This leads to a quadratic system inthe unknown entries of q1 and q2. This system is already very complex for MatLab.Nevertheless, one of the possible solutions is shown in Figure 2.1.

r q1, q2 s “

»

—

—

—

—

—

—

—

—

–

´0.1134 ´0.6343´0.4535 1.0460

0.8658 ´0.2263´0.0554 ´0.1484´0.1287 ´0.0371´0.0919 0´0.0230 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

-1 0 1 2 3 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 0 1 2 3-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 2.1: Two possible generators for the cubic B-spline regular wavelet tight frameover the initial mesh Z.

If we take a look back to the structure of Rj we can try to infer more dependency tosimplify the quadratic system. First we observe that the upper right and lower left 2ˆ2corners in red (for the interpretation of the references to color, the reader is referred tothe pdf version of this manuscript) should belong to only one block Bp¨ ´ 2k, ¨ ´ 2kq.Moreover, the numbers in blue, in magenta and cyan are sum of the entries of two,three and four consecutive blocks, respectively. Since we cannot have qm with only oneelement (the corresponding framelet would be a multiple of a scaling function which hasno vanishing moments), we can guess

Rj “ÿ

kZ

B7p¨ ´ 2k, ¨ ´ 2kq ` B5p¨ ´ 2k, ¨ ´ 2kq ` B3p¨ ´ 2k, ¨ ´ 2kq,

with Bn P Rnˆn being rank-1 blocks. Moreover, due to the bi-symmetry of Rj wesuppose

B7 “ q1 qT1 , B5 “ q2 qT2 , B3 “ q3 qT3 ,

with symmetric tqmum“1,2,3. This simplifies enormously the resulting quadratic systemand leads to a unique solution with symmetric generators, Figure 2.2. The price for ourassumptions is that only one generator has a small support.

Semi-regular Case

Consider now the initial semi-regular mesh t0 in (1.3) with h` “ 1 and hr “ 2. Similarlyto Example 1.32, we get a subdivision matrix P that differs from the regular one overthree columns, w.l.o.g. k`pPq “ ´2 and krpPq “ 2. Thus, we have three irregular basiclimit functions. To construct the wavelet tight frame, first of all we need to renormalizethe basic limit functions and the subdivision matrix as in (2.5) and (2.6). To do so we

54


r q1, q2, q3 s “»

—

—

—

—

—

—

—

—

—

–

?6{48

?22{48

?39{18

?6{12

?22{12 ´

?39{9

5?

6{144 ´5?

22{24?

39{18

´5?

6{18?

22{12 0

5?

6{144?

22{48 0?

6{12 0 0?

6{48 0 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

-1 0 1 2 3 4-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 0 1 2 3-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-1 0 1 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Figure 2.2: Three possible symmetric generators for the cubic B-spline regular wavelettight frame over the initial mesh Z.

compute the integrals of the basic limit functions tϕkukPZ. From the regular case, wealready know that

ż

Rϕkpxq dx “

$

&

%

1, if k ď k`pPq,

2, if k ě krpPq.

To compute the missing integrals we rely on Proposition 1.40. Thus,

»

—

—

—

—

—

—

—

—

—

—

—

—

–

1{81{2

25{32 3{325{8 3{8

5{24 29{40 1{153{5 2{5

3{20 29{401{21{8

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

T

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

111

ż

Rϕ´1pxq dx

ż

Rϕ0pxq dx

ż

Rϕ1pxq dx

222

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

“ 2

»

—

—

—

—

—

—

—

—

—

–

ż

Rϕ´1pxq dx

ż

Rϕ0pxq dx

ż

Rϕ1pxq dx

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

which leads toż

Rϕ´1pxq dx “

5

4,

ż

Rϕ0pxq dx “

3

2,

ż

Rϕ1pxq dx “

7

4.

55


Thus,

Pj “1?

2

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

1{81{2

3{4?

5{20. . . 1{2

?5{5

1{8 5?

5{16?

6{32

5{8?

30{16?

30{24 29{40?

42{105?

42{10 2{5?

3{5 29?

14{140 1{8?

14{7 1{2. . .

?14{28 3{4

1{21{8

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

The next step is to find suitable matrices Sj. Due to what we know from the regularcase and the symmetry of Sj we are in the following situation

Sj “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

. . . . . .

. . . 5{3 ´1{3´1{3 ¨ ¨ ¨ ¨ ¨

¨ ¨ ¨ ¨ ¨

¨ ¨ Sirr ¨ ¨

¨ ¨ ¨ ¨ ¨

¨ ¨ ¨ ¨ ¨ ´1{3

´1{3 5{3. . .

. . . . . .

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

with Sirr P R5ˆ5 symmetric. At this point, any Sj such that

Sj mj,α “ cj,α, α “ 0, 1, (2.24)

and Sj´Pj´1Sj´1PTj´1 is positive semi-definite, is fine because the problem of exchanging

integrals in Theorem 2.15 appears only at the boundaries and we only changed locally afinite number of compactly supported scaling functions and a finite section of Sj. Dueto the renormalization we have

mj,0 “ cj,0 ““

. . . , 1,?

5{2,?

6{2,?

7{2,?

2, . . .‰T, j P N0.

Similarly to the regular case, the vector cj,1 of the coefficients that generates the mono-mial x belongs to the right-eigenspace of P associated to the eigenvalue 1{2. In partic-

56


ular, see Example 1.32,

cj,1 “ 2´32j D1{2

»

—

—

—

—

—

—

—

—

—

–

...´2´11{3

24...

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

“ 2´32j

$

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

%

k, k ď ´2,

´

?5

2, k “ ´1,

?6

6, k “ 0,

?7, k “ 1,

2?

2k, k ě 2,

, j P N0.

To compute mj,1 we use again Proposition 1.40. From the regular case, together with(1.32), we have

ż

Rϕkpxq dx “

$

&

%

k, if k ď k`pPq “ ´2,

4k, if k ě krpPq “ 2,

thus,

»

—

—

—

—

—

—

—

—

—

—

—

—

–

1{81{2

25{32 3{325{8 3{8

5{24 29{40 1{153{5 2{5

3{20 29{401{21{8

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

T

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

´4´3´2

ż

Rx ϕ´1pxq dx

ż

Rx ϕ0pxq dx

ż

Rϕ1pxq dx

81216

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

“ 4

»

—

—

—

—

—

—

—

—

—

–

ż

Rx ϕ´1pxq dx

ż

Rx ϕ0pxq dx

ż

Rx ϕ1pxq dx

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

,

which leads toż

Rx ϕ´1pxq dx “ ´1,

ż

Rx ϕ0pxq dx “

9

10,

ż

Rx ϕ1pxq dx “

77

20.

57


Therefore,

mj,1pkq “ 2´32j

$

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

%

k, k ď ´2,

´2?

5

5, k “ ´1,

3?

6

10, k “ 0,

11?

7

10, k “ 1,

2?

2k, k ě 2.

If, similarly to the regular case, we suppose Sirr to be tridiagonal, then the condition(2.24) yields a unique solution

Sirr “

»

—

—

—

—

—

–

29{18 ´?

5{9

´?

5{9 14{9 ´?

30{18

´?

30{18 27/16 ´59?

42{1008

´59?

42{1008 2683{1512 ´20?

14{189

´20?

14{189 46{27

fi

ffi

ffi

ffi

ffi

ffi

fl

,

which leads to a positive semi-definite Sj´Pj´1Sj´1PTj´1. Indeed, the changes in Pj and

Sj affect the columns of Rj with indices ´6 to 6, Figure 2.3. If we subtract the regularblocks we found before, e.g. in Figure 2.2, from both sides until we eliminate all theentries which did not change from the regular to the semi-regular case, we end up withthe 13ˆ 13 block Rirr in Figure 2.4. Rirr is positive semi-definite and can be factorizedas QirrQ

Tirr in different ways. For example, we can use the eigenvalue decomposition,

Figure 2.5. This way we obtain nine frame elements around t0p0q which all have supportin r´3, 6s. This choice however spoils completely the blocking structure still visible inRirr. Another possible choice to preserve this block structure is to use a Cholesky-likefactorization. This is what have been done in Figure 2.6. Here the Cholesky algorithmhas been applied alternatively from the left and from the right side, not to promoteeither of the directions approaching t0p0q. This approach results in a much sparser Qirr

and in framelets with much shorter support. In Figure 2.6, one can observe also thatthe framelets obtained this way have a more uniform behaviour and are less oscillatingthan the ones in Figure 2.5. The same process can be done starting from the regularframelets in 2.1, with similar results. This example shows that already in a fairly simplecase the matrices involved are messy and difficult to handle. Studying the whole processwith respect to the initial mesh parameters h`, hr then becomes a nightmare, apart fromvery special cases. In the next section, we will construct a family of wavelet tight framesfor which a lot of the steps presented here are much simpler. The trade-off is a toll withrespect to the possible choices of h` and hr.

58


»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

1

384

1

96

7

384

1

96

1

384

1

48

1

24

1

96

´7

128

1

48

7

384

1

96

1

384

´59

96´

1

8

1

48

1

24

1

96

2845

2304´

353

576´

61

1152

11

576

1

72

5?

5

1152

5?

6

3456

´353

576

193

144´

175

288´

19

144

1

288

5?

5

288

5?

6

864

´61

1152´

175

288

7159

5760´

443

720´

151

2304

5?

5

1152

97?

6

17280

?7

240

?2

480

11

576´

19

144´

443

720

983

720´

329

576´

5?

5

72´

7?

6

1080

?7

60

?2

120

1

72

1

288´

151

2304´

329

576

184583

147456´

18361?

5

73728´

54791?

6

1548288

89 ˚?

7

7168

281?

2

21504

59?

2

10752

59?

2

43008

5?

5

1152

5?

5

288

5?

5

1152´

5?

5

72´

18361?

5

73728

44899

36864´

76295?

30

774144´

181?

35

10752

19?

10

3584

59?

10

5376

59?

10

21504

5?

6

3456

5?

6

864

97?

6

17280´

7?

6

1080´

54791?

6

1548288´

76295?

30

774144

51511849

40642560´

1117199?

42

10160640´

94597?

3

5080320

14765?

3

508032

23725?

3

2032128

2?

3

567

?3

1134

?7

240

?7

60

89?

7

7168´

181?

35

10752´

1117199?

42

10160640

83249

60480´

73253?

14

423360´

955?

14

42336

1285?

14

169344

2?

14

189

?14

378

?2

480

?2

120

281?

2

21504

19?

10

3584´

94597?

3

5080320´

73253?

14

423360

517087

423360´

1711

2646´

7853

169344

205

6048

65

3024

59?

2

10752

59?

10

5376

14765?

3

508032´

955?

14

42336´

1711

2646

13823

10584´

12979

21168´

179

1512

17

756

59?

2

43008

59?

10

21504

23725?

3

2032128

1285?

14

169344´

7853

169344´

12979

21168

52055

42336´

1871

3024´

337

6048

2?

3

567

2?

14

189

205

6048´

179

1512´

1871

3024

287

216´

133

216

?3

1134

?14

378

65

3024

17

756´

337

6048´

133

216

4265

3456

1

96

1

24

1

48´

1

8´

59

96

1

384

1

96

7

384

1

48´

7

128

1

96

1

24

1

48

1

384

1

96

7

384

1

96

1

384

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

Figure 2.3: The columns of Rj that differ from the regular case.

59


» — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — –

919

6912

´329

1728

107

3456

1

576

11

1152

5?

5

1152

5?

6

3456

´329

1728

295

432

´517

864

1

144

11

288

5?

5

288

5?

6

864

107

3456

´517

864

10571

8640

´911

1440

´161

2304

5?

5

1152

97?

6

17280

?7

240

?2

480

1

576

1

144

´911

1440

953

720

´335

576

´5?

5

72

´7?

6

1080

?7

60

?2

120

11

1152

11

288

´161

2304

´335

576

184199

147456

´18361?

5

73728

´54791?

6

1548288

89?

7

7168

281?

2

21504

59?

2

10752

59?

2

43008

5?

5

1152

5?

5

288

5?

5

1152

´5?

5

72

´18361?

5

73728

44899

36864

´76295?

30

774144

´181?

35

10752

19?

10

3584

59?

10

5376

59?

10

21504

5?

6

3456

5?

6

864

97?

6

17280

´7?

6

1080

´54791?

6

1548288

´76295?

30

774144

51511849

40642560

´1117199?

42

10160640

´94597?

3

5080320

14765?

3

508032

23725?

3

2032128

2?

3

567

?3

1134

?7

240

?7

60

89?

7

7168

´181?

35

10752

´1117199?

42

10160640

83249

60480

´73253?

14

423360

´955?

14

42336

1285?

14

169344

2?

14

189

?14

378

?2

480

?2

120

281?

2

21504

19?

10

3584

´94597?

3

5080320

´73253?

14

423360

230491

211680

´1069

2352

´6401

84672

23

756

23

3024

59?

2

10752

59?

10

5376

14765?

3

508032

´955?

14

42336

´1069

2352

247

392

´331

42336

´25

189

´25

756

59?

2

43008

59?

10

21504

23725?

3

2032128

1285?

14

169344

´6401

84672

´331

42336

823

56448

5

378

5

1512

2?

3

567

2?

14

189

23

756

´25

189

5

378

1 27

1

108

?3

1134

?14

378

23

3024

´25

756

5

1512

1

108

1

432

fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl

Figure 2.4: The remainder part Rirr of Rj once the regular blocks are subtracted.

60


Qirr “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

´0.0068 ´0.0411 0.0395 ´0.0580 0.0799 ´0.1228 0.2294 0.2075 0.0956

0.0504 0.3056 ´0.2687 0.3645 ´0.4198 0.3503 ´0.2792 ´0.0481 0.0509

´0.1399 ´0.7702 0.5243 ´0.4412 0.2929 0.0171 ´0.2003 ´0.1212 0.0191

0.2198 0.9527 ´0.2021 ´0.2285 0.4223 ´0.2789 ´0.0090 ´0.1354 ´0.0074

´0.2441 ´0.6365 ´0.6252 0.5174 ´0.0646 ´0.2979 0.1377 ´0.1150 ´0.0277

0.4285 0.1531 0.8435 0.1903 ´0.4615 ´0.0239 0.2088 ´0.0622 ´0.0461

´0.8627 0.1755 ´0.2397 ´0.5319 ´0.1743 0.3067 0.1541 0.0248 ´0.0566

0.9804 ´0.3061 ´0.3794 ´0.0930 0.2726 0.2852 0.0060 0.1007 ´0.0559

´0.5258 0.1894 0.3966 0.6212 0.4387 0.0001 ´0.1479 0.1318 ´0.0409

0.0748 ´0.0327 ´0.1107 ´0.3454 ´0.4655 ´0.4447 ´0.2495 0.1216 ´0.0199

0.0231 ´0.0088 ´0.0209 ´0.0370 ´0.0243 0.0274 0.0584 ´0.0697 0.0506

0.0035 ´0.0005 0.0055 0.0395 0.0737 0.1108 0.0966 ´0.0801 0.0445

0.0009 ´0.0001 0.0014 0.0099 0.0184 0.0277 0.0242 ´0.0200 0.0111

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

-5 0 5 10-0.5

0

0.5

-5 0 5 10-1

0

1

-5 0 5 10-1

0

1

-5 0 5 10-1

0

1

-5 0 5 10-0.5

0

0.5

-5 0 5 10-0.5

0

0.5

-5 0 5 10-0.5

0

0.5

-5 0 5 10-0.2

0

0.2

-5 0 5 10-0.2

0

0.2

Figure 2.5: The result of the eigenvalue decomposition of Rirr and the correspondingframelets.

61


Qirr “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

0.3646

´0.5222 0.6405

0.0849 ´0.8650 0.6841

0.0048 0.0147 ´0.9067 0.7080

0.0262 0.0810 ´0.0030 ´0.8272 0.7401 0.0975 0.0195

0.0266 0.0823 0.1150 ´0.0740 ´0.8693 0.6320 0.1703 0.0874

0.0097 0.0301 0.0569 0.0498 ´0.0983 ´1.0817 0.2167 0.1817 0.0317

0.0161 0.0829 0.2662 0.4673 ´1.0085 0.1436 0.2057

0.0043 0.0222 0.5521 ´0.8709 0.1581

0.3971 ´0.6873

0.0993 0.0687

0.1925

0.0481

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

-4 -2 0 2-0.4

-0.2

0

0.2

0.4

-4 -2 0 2-1

-0.5

0

0.5

-4 -2 0 2 4-1

-0.5

0

0.5

1

-4 -2 0 2 4-1

-0.5

0

0.5

1

-2 -1 0 1 2 3-1

-0.5

0

0.5

1

-2 -1 0 1 2 3-1

-0.5

0

0.5

-2 0 2 4-1

-0.5

0

0.5

-2 0 2 4 6-0.5

0

0.5

-2 0 2 4 6 8-0.6

-0.4

-0.2

0

0.2

Figure 2.6: The result of the alternating application of the Cholesky factorization of Rirr

and the corresponding framelets.

62


2.2 Semi-regular Dubuc-Deslauriers Wavelet TightFrames

In this section, we present the work published in [60]. The aim is to provide an easystrategy to construct a semi-regular family of wavelet tight frames with a high numberof vanishing moments. The starting point for this construction is the family of Dubuc-Deslauriers subdivision schemes, introduced in [27] in the regular case and extended tothe semi-regular setting in [62]. These schemes are interpolatory, i.e. their basic limitfunctions tϕkukPZ satisfy

ϕkpt0pmqq “ δkm, k,m P Z, (2.25)

where t0 is the initial mesh in (1.3). Moreover, the Dubuc-Deslauriers family depends ona parameter n P N, that describes the polynomial generation of each scheme. Indeed, theDubuc-Deslauriers schemes can be constructed as solutions of the following interpolationproblems, which already incorporates the semi-regular setting.

Definition 2.23. Let n P N and t0 a semi-regular initial mesh in (1.3). The subdivisionmatrix P defining the Dubuc-Deslauriers 2n-point scheme over t0 is the unique bi-infinitematrix satisfying the following conditions:

1. Pp2k, kq “ 1, k P Z,

2. the entries Pk “ rPp2k ` 1,mq : m “ k ´ n` 1, . . . , k ` ns, k P Z, satisfy

Pk

»

–

1 t0pk ´ n` 1q ¨ ¨ ¨ t0pk ´ n` 1q2n´1

1 t0pk ´ n` 2q ¨ ¨ ¨ t0pk ´ n` 2q2n´1

......

. . ....

1 t0pk ` nq ¨ ¨ ¨ t0pk ` nq2n´1

fi

fl “

”

1t0p2k ` 1q

2¨ ¨ ¨

ˆ

t0p2k ` 1q

2

˙2n´1ı

.

3. all other entries of P are equal to zero.

Remark 2.24. Definition 2.23 is well posed since the linear systems in part 2. of Definition2.23 are uniquely solvable due to t0 being monotonically increasing. Moreover, for everyk P Z,

supppPp:, kqq “ t2k ´ 2n` 1, . . . , 2k ` 2n´ 1u

which means that the columns of P have support of length 4n´ 1 and that the eventualbasic limit functions have

supppϕkq “ rt0p2k ´ 2n` 1q, t0p2k ` 2n´ 1qs. (2.26)

3

Here we present a proof, for any mesh t0, with arbitrary h`, hr P p0,8q, of the con-vergence of these schemes in the semi-regular setting.

Proposition 2.25. Let n P N and P be the subdivision matrix constructed in 1.-3. overthe semi-regular mesh t0. Then

63


(i) 1 is a simple eigenvalue of P associated to the right eigenvector 1 and all othereigenvalues of P are less than 1 in absolute value;

(ii) the subdivision scheme with the subdivision matrix P converges.

Proof. Part piq: Let n P N and I “ t1 ´ 2n, . . . , 2n ´ 1u. By Proposition 1.9, all non-zero eigenvalues of P are uniquely determined by the eigenvalues of its finite section, thesquare matrix P “ pPpm, kqqm,kPI . By construction, due to 2., P1 “ 1. We show next,

that λ P Czt0, 1u with Pv “ λv, v P C4n´3zt0u, must satisfy |λ| ă 1. The proof is by

contradiction, we assume that |λ| ě 1. Note first that Pp0, 0q “ 1 and by step 3. of theconstruction above, we get vp0q “ λ vp0q, thus, vp0q “ 0. Step 1. of the constructionforces vpkq “ λ vp2kq for k, 2k P I. To determine the odd entries of v, let m P Ibe odd and consider the polynomial interpolation problem with the pairwise distinct

knots t0pkq and values vpkq for j P

"

m` 1

2´ n, . . . , n`

m` 1

2

*

. This interpolation

problem possesses a unique solution, possibly complex-valued, interpolation polynomialπ P Π2n´1. Therefore, by the interpolation property of P, we have

λ π pt0pmqq “ λ vpmq “´

P v¯

pmq “ π

ˆ

t0pmq

2

˙

.

Iterating we obtain

limrÑ8

|λ|r |π pt0pmqq | “ limrÑ8

ˇ

ˇ

ˇ

ˇ

π

ˆ

t0pmq

2r

˙ˇ

ˇ

ˇ

ˇ

“ |πp0q| “ |vp0q| “ 0, (2.27)

which leads to a contradiction: for |λ| “ 1, we have vpmq “ πpt0pmqq “ 0 or, for|λ| ą 1, the identity (2.27) is violated. It is left to show that λ “ 1 is simple. The proofis by contradiction. w.l.o.g. we assume that 1 has an algebraic multiplicity 2. Note thatδT P “ δT , where δp0q “ 1 and its other entries are equal to zero and define A “ P´1δT .Then A has a simple eigenvalue 1 with A v “ v, v ‰ 0. By construction vp0q “ 0, thusAv “ Pv. Following a similar argument as above we arrive at the contradiction v “ 0.Part piiq: To prove the convergence of the scheme, due to [27, 62], it suffices to provethe continuity of the basic limit functions at 0, i.e.

|fjp0q ´ fjp1q| ÝÑjÑ8

0, and |fjp0q ´ fjp´1q| ÝÑjÑ8

0 (2.28)

for fj “ Pj epkq, k P Z. Equivalently, we show that

|fjp0q ´ fjp1q| “ˇ

ˇ r 0, . . . , 0, 1 ,´1, 0, . . . , 0 s Pj epkqˇ

ˇÑ 0

and, similarly, for the other difference in (2.28). The claim follows then by part piq,together with steps 1. and 3. of the construction, which imply that Pj epkq Ñ epkqp0q 1.

Due to 1. these schemes preserves at each level all the data computed at the previous

64


levels. In particular, for every initial data f0 P `pZq,

fjp2kq “ pPfj´1q pkq, j P N, k P Z,

and this is sufficient to guarantee (2.25). The odd entries of each finer level instead arecomputed using linear combinations of the 2n neighbouring points. Moreover, condition2. of Definition 2.23 implies that the constructed schemes reproduces polynomials ofdegree 2n´ 1, see Definition 1.18.

Remark 2.26. A subdivision scheme over the initial mesh t0 that reproduces polynomialsof degree v P N has a subdivision matrix P that satisfies, for every polynomial π P Πv

of degree v,πptjq “ P πptj´1q, k P Z.

3

The convergence and the interpolation property imply that the functions tϕk : k P Zuare linearly independent and, thus, the representation

xα “ÿ

kPZ

t0pkqα ϕkpxq , x P R, α P t0, . . . , 2n´ 1u, (2.29)

is unique. If we suppose that the integrals of tϕkukPZ are positive, the correspondingscaling functions Φj “ rφj,k : k P Zs in (2.5) inherit the polynomial reproductionproperty in (2.29) and we have

xα “ ΦTj pxq cj,α, cj,α “ 2´

32j D1{2 tα0 , α P t0, . . . , 2n´ 1u. (2.30)

Furthermore, in the semi-regular case, we have k`pPq “ 1 ´ 2n and krpPq “ 2n ´ 1,which means the presence of 4n´ 3 irregular basic limit functions corresponding to theindices

Iirr “ t2´ 2n, . . . , 2n´ 2u. (2.31)

We denote with Φ` and Φr the vectors of scaling functions over the regular meshest` “ h`Z and tr “ hrZ respectively, and thus

I` Φ “ I` Φ` and Ir Φ “ Ir Φr, (2.32)

where

I`pm, kq “

$

&

%

1, if m “ k ă 2´ 2n,

0, otherwise,and Irpm, kq “

$

&

%

1, if m “ k ą 2n´ 2

0, otherwise.

The choice of the Dubuc-Deslauriers schemes is due to the fact that they are one ofthe special cases in which the frames constructed via the UEP conditions, (2.8) withspωq ” 1, achieve naturally more than one vanishing moment, and this can be exploitedto simplify the construction in the semi-regular case.

65


2.2.1 Wavelet Tight Frames Construction

We present the construction of the wavelet tight frames based on the Dubuc-Deslauriers2n-point schemes as we did for the cubic B-spline scheme in Section 2.1.1, starting withthe regular case, where we will exploit Theorem 2.9, and then passing to the semi-regularcase with an ad hoc construction. Unfortunately, as it will be shown in Section 2.2.2,the construction is not possible for every choice of h`, hr ą 0, but the larger n is thesmaller the interval which the ratio h`{hr can belong, see Section 2.2.2, case n ą 2.

Regular Case

The construction in the regular case is based on another family of schemes leading tothe Daubechies 2n-tap wavelet systems (see e.g. [20]), which form orthonormal basis forL2pRq. In particular, for n P N, the Daubechies 2n-tap schemes are characterized by the

unique symbol dpωq with coefficients d supported on t1´2n, . . . , 0u which satisfies (2.8)with spωq ” 1 and

qdpωq “ q1pωq “ e´i2πp2n´1qω dpω ´ 1{2q and v “ n, n P N. (2.33)

Daubechies wavelets are closely connected to the Dubuc-Deslauriers subdivision schemes.Indeed, see [51], the symbol ppωq of the Dubuc-Deslauriers 2n-point scheme satisfies

ppωq “ dpωq dpωq, ω P R. (2.34)

The identity (2.34), together with Theorem 2.9, leads to our construction of Dubuc-Deslauriers wavelet tight frames for the regular case.

Proposition 2.27. Let n P N and dpωq and ppωq be the symbols of the Daubechies2n-tap scheme and the Dubuc-Deslauriers 2n-point scheme, respectively. Then

q1pωq “?

2 ei2πp2n´1qωdpωq dpω ´ 1{2q

q2pωq “ dpω ´ 1{2q dpω ´ 1{2q,

ω P R,

define a wavelet tight frame with n vanishing moments for ppωq in Theorem 2.8.

Proof. Note that the convergence of the subdivision associated to dpωq implies the con-vergence of the subdivision associated to dpωq. Thus, applying Theorem 2.9 to dpωqand dpωq, due to (2.34), we obtain a wavelet tight frame for ppωq “ dpωqdpωq with nvanishing moments with the polynomials

dpωq qdpωq, dpωq qdpωq, and qdpωq qdpωq.

To reduce the number of frame generators we take a closer look to the structure of these

66


framelets. Indeed, the UEP identities, (2.8) with spωq ” 1, and (2.33) yield

$

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

’

%

1 “

´

dpωq dpωq ` qdpωq qdpωq¯ ´

dpωq dpωq ` qdpωq qdpωq¯

“

´

dpωq dpωq¯ ´

dpωq dpωq¯

`?

2 dpωq qdpωq´?

2 dpωq qdpωq¯

`

´

qdpωq qdpωq¯ ´

qdpωq qdpωq¯

,

0 “

´

dpωq dpω ´ 1{2q ` qdpωq qdpω ´ 1{2q¯ ´

dpωq dpω ´ 1{2q ` qdpωq qdpω ´ 1{2q¯

“

´

dpωq dpωq¯ ´

dpω ´ 1{2q dpω ´ 1{2q¯

`

´

qdpωq qdpωq¯ ´

qdpω ´ 1{2q qdpω ´ 1{2q¯

` dpωq qdpωq´

dpω ´ 1{2q qdpω ´ 1{2q¯

` dpωq qdpωq dpω ´ 1{2q qdpω ´ 1{2q.

(2.35)

From (2.33), we have

qdpωq dpω ´ 1{2q “ qdpωq dpω ´ 1{2q, ω P R.

Moreover, the periodicity of the symbols implies

dpωq qdpω ´ 1{2q “ dpωq qdpω ´ 1{2q, ω P R.

Next, we rewrite the last term of the second identity in (2.35)

dpωq qdpωq dpω ´ 1{2q qdpω ´ 1{2q “ dpωq qdpωq´

dpω ´ 1{2q qdpω ´ 1{2q¯

, ω P R,

obtaining q1pωq “?

2dpωqqdpωq and q2pωq “ qdpωqqdpωq. The claim follows by (2.33).

Remark 2.28. This construction leads to the same result as in [12, Section 3.1.2], but ina more straightforward way. 3

An important consequence of Proposition 2.27 is that from the matrix point of viewSj ” I gives n vanishing moments in Theorem 2.15. Similarly to the construction inSection 2.1.1 in the semi-regular case, we exploit the regular wavelet tight frame and thefact that Sj ” I to isolate the irregular part of the matrix in Theorem 2.15 (iii).

Semi-regular Case

Let n P N, we consider the semi-regular Dubuc-Deslauriers 2n-point scheme over thesemi-regular initial mesh t0 in (1.3) with h`, hr ą 0. To apply Theorem 2.15, we needto check first Assumptions 1, 2 and 3. Assumption 1 follows from Proposition 2.21due to the linear independence of the scaling functions guaranteed by the interpolationproperty (2.25). Assumption 2 is a direct consequence of the polynomial reproduction(2.30) and the third one is satisfied, as long as the integrals of the basic limit functionsare positive, due to (2.19) and [22].

The next step, for the construction of the wavelet tight frames is the choice of thematrices tSjujPN, j P N. A natural choice, at least to get one vanishing moment, should

67


be Sj ” I. After all, we always have cj,0 “ mj,0, j P N0. Unfortunately, already for the4-point scheme: the matrix I´PjP

Tj is not positive semi-definite. The idea is to change

the matrix Sj locally nearby the irregular scaling functions, i.e. we have the followingsituation

Sj “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

. . .

11¨ ¨ ¨ ¨ ¨

¨ ¨ ¨ ¨ ¨

¨ ¨ Sirr ¨ ¨

¨ ¨ ¨ ¨ ¨

¨ ¨ ¨ ¨ ¨

11

. . .

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

with Sirr P Rp4n´3qˆp4n´3q. Since in the regular case we obtain n vanishing moments, Sirrmust be chosen to maintain this property. To achieve this goal, since Sj must satisfy(2.17), we need to know the vectors cj,α and mj,α, for every α P t0, . . . , 2n ´ 1u. Thisis not difficult thanks to the polynomial reproduction property (2.30) and Proposition1.40. Since outside Sirr, the matrices Sj are diagonal, there is no interaction betweenregular and irregular entries. The idea, roughly speaking, is to choose Sirr such that it isthe minimal block that achieves (2.24) to get the desired number of vanishing moments.

Algorithm 1:

1. Define the p4n´ 3q ˆ n matrix

C ““

rcj,0pkqskPIirr | . . . | rcj,n´1pkqskPIirr‰

;

2. Compute the QR factorization C “ OU with orthogonal O P Rp4n´3qˆp4n´3q andupper triangular U P Rp4n´3qˆn;

3. Define Sj ” I and, if h` ‰ hr, modify

Sirr :“ rSjpk,mqsk,mPIirr “rO rOT , (2.36)

whererO “

“

rOpk, 1qskPIirr | . . . | rOpk, nqskPIirr‰

.

We notice that Algorithm 1, when h` ‰ hr, generates a matrix Sirr which is not fullrank and this is a significant downside for some applications, e.g. signal compression.Nonetheless, for analysis of subdivision smoothness we only need the decomposition partof the corresponding wavelet tight frame algorithm and the matrices Sj constructed viaAlgorithm 1 define a family of kernels Φjpxq

TSjΦjpyq with the desired approximationproperties.

68


Proposition 2.29. Let Sj be defined by Algorithm 1. Then, for all f P L2pRq,

piq DCj ą 0 :

ż

R2

fpxq ΦjpxqT Sj Φjpyq fpyq dx dy ď Cj }f}

2L2,

piiq

ż

R2

fpxq ΦjpxqT Sj Φjpyq fpyq dx dy ÝÑ }f}2L2 , j Ñ 8.

Proof. Part piq: Since Φj “ 2j{2Φ0p2j¨q, we only need to prove the claim for j “ 0.

Recall that the regular elements in Φ0 are the elements of Φ` and Φr in (2.32). For suchregular families of scaling functions Theorem 2.15 with S “ I implies the existence ofC` ą 0 and Cr ą 0, respectively, such that, for all f P L2

pRq,

max

"ż

R2

fpxq Φ`pxqT Φ`pyq fpyq dx dy,

ż

R2

fpxq ΦrpxqT Φrpyq fpyq dx dy

*

ď

ď maxtC`, Cru }f}2L2 .

Decompose the bi-infinite identity I “ I` ` Iirr ` Ir with

I`pj, jq “

"

1, j ă 1´ 2n,0, otherwise,

and Irpj, jq “

"

1, j ą 2n´ 1,0, otherwise.

Then, for all f P L2pRq, by (2.36) and the Cauchy-Schwarz inequality, we have

ż

R2

fpxq Φ0pxqT S0 Φ0pyq fpyq dx dy “

“

ż

R2

fpxq ΦpxqT0 p I` ` Iirr S0 Iirr ` Ir q Φpyq0 fpyq dx dy

ď maxtC`, Cru }f}2L2 `

2n´2ÿ

j“2´2n

2n´2ÿ

k“2´2n

|S0pj, kq|

ˇ

ˇ

ˇ

ˇ

ż

Rfpxq φ0,jpxq dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ż

Rφ0,kpyq fpyq dy

ˇ

ˇ

ˇ

ˇ

ď

ˆ

maxtC`, Cru ` p4n´ 3q }Sirr}8 maxkPIirr

}φ0,k}2L2

˙

}f}2L2 .

69


Part piiq: Using again I “ I` ` Iirr ` Ir and (2.26), for every f P L2pRq, we get

›

›

›

›

f ´

ż

Rfpxq Φjpxq

T Sj Φjp¨q dx

›

›

›

›

2

L2

ď

›

›

›

›

f χp´8,0q ´ 2jż 0

´8

fpxq Φ`p2jxqT Φ`p2

j¨q dx

›

›

›

›

2

L2

`

›

›

›

›

2jż

Rfpxq Φp2jxqT Iirr S0 Iirr Φp2j¨q dx

›

›

›

›

2

L2

`

›

›

›

›

f χp0,8q ´ 2jż 8

0

fpxq Φrp2jxqT Φrp2

j¨q dx

›

›

›

›

2

L2

“: γ` ` γirr ` γr.

The indices of the non-zero elements of Iirr S0 Iirr belong to the set Iirr ˆ Iirr in (2.31),thus,

ď

kPIirr

supppφ0,kq “: ra, bs, ´8 ă a ď b ă 8.

The continuity of Φ0 and the Cauchy-Schwarz inequality yield

γirr :“ 22j

ż

ra,bs

2j

ˇ

ˇ

ˇ

ˇ

ˇ

ż

ra,bs

2j

fpxq Φ0p2jxqT Iirr S Iirr Φ0p2

jyq dx

ˇ

ˇ

ˇ

ˇ

ˇ

2

dy ď C }f}2L2p

ra,bs

2jq

with the constant C “ pb ´ aq2}ΦT Iirr S0 Iirr Φ}2L8 . Thus, γirr goes to zero as j goesto 8. Moreover, since fχp´8,0q and fχp0,8q belong to L2

pRq, by the argument from theregular case, both γ` and γr go to zero as j goes to 8.

Examples in Section 2.2.2 and numerical evidence for n “ 3, . . . , 8 with different h`, hr ą0 (defining the mesh t0) lead to the following conjecture.

Conjecture 2.30. Let Sj be defined by Algorithm 1, Pj and D as in (2.6) for theDubuc-Deslauriers 2n-point scheme and p, q1 and q2 as in Proposition 2.27.

piq ForRj “ Sj ´ Pj´1 Sj´1 PT

j´1, (2.37)

and Bk, k R Iirr with entries

Bkpt, uq “ ppT pt´ 2k, u´ 2kq `2ÿ

m“1

qmqTmpt´ 2k, u´ 2kq, t, u P Z, (2.38)

the matrix

Rirr “ Rj ´1

2

ÿ

kRIirr

Bk (2.39)

is positive semi-definite.

70


piiq For all α P t0, . . . , n´ 1u, Sj satisfy (2.24).

Remark 2.31. Note that the requirement that Rirr is positive semi-definite is strongerthan the positive semi-definiteness of Rj. Moreover, we get explicit

Qj ““

. . . QminpIirrq´1 Qirr QmaxpIirrq`1 . . .‰

with Rirr “ QirrQTirr and Bk “ QkQ

T

k for k R Iirr. 3

Apart from the examples in Section 2.2.2 and numerical evidence for n “ 3, . . . , 8,there are other facts that support Conjecture 2.30. First of all, in the regular case,the construction in Algorithm 1 reduces to the standard UEP construction. Indeed,even if we modify Sirr as in Algorithm 1 step 3, since in the regular case mj,α “ cj,α,α P t0, . . . , n´ 1u, it is easy to check that Conjecture (i) and (ii) are satisfied. Secondly,in the semi-regular case, (2.24) reduces to an identity for certain finite matrices. Forα “ 0, . . . , n´ 1, define

Mj ““

rmj,0pkqskPIirr | . . . | rmj,n´1pkqskPIirr‰

andCj “

“

rcj,0pkqskPIirr | . . . | rcj,n´1pkqskPIirr‰

.

Then the irregular part of (2.24) becomes SirrMj “ Cj, which implies

MTj Cj “ MT

j Sirr Mj.

Thus, for (2.24) to hold the matrix MTj Cj must be symmetric. Indeed, for every

α, β P t0, . . . , n´ 1u, by (2.30), we get

0 “ xαΦTj cj,β ´ cTj,α Φjx

β

“ rxαφj,kpxqsTkPIirr rcj,βpkqskPIirr ´ rcj,αpkqs

TkPIirr rx

βφj,kpxqskPIirr

`ÿ

kRIirr

`

cj,βpkq xαφj,kpxq ´ cj,αpkqx

βφj,kpxq˘

, x P R.

Integrating both sides of the above identity and using the fact that mj,αpkq “ cj,αpkqfor α P t0, . . . , n´ 1u, k R Iirr, we obtain

pMTj Cjqpα, βq ´ pM

Tj Cjqpβ, αq “

“ rmj,αpkqsTkPIirr rcj,βpkqskPIirr ´ rcj,αpkqs

TkPIirr rmj,βpkqskPIirr

“ 0, α, β P t0, . . . , n´ 1u.

We strongly believe that part piiq of Conjecture 2.30 is due to some special, intriguingproperty of the Dubuc-Deslauriers schemes.

71


Remark 2.32. If one chooses rn ă n columns of O in (2.36), then the correspondingmatrix S would generate a wavelet tight frame with rn vanishing moments. Since it isnot possible to get more than n vanishing moments in the regular case, the choice rn “ nis optimal in the semi-regular case. 3

2.2.2 Examples

We present two simple examples illustrating the construction in Section 2.2 for n “ 1and n “ 2, respectively. The small bandwidth of the corresponding subdivision matricesP allows for exact computations in terms of the mesh parameter h`, hr ą 0. Withoutloss of generality, after a suitable renormalization, we consider the semi-regular mesh t0

with h` “ 1 and hr “ h, h ą 0. In the case n “ 1, the Dubuc-Deslauriers 2-point schemecorresponds to the linear B-spline scheme, Examples 1.5, 1.12, 1.20 and 1.25. The casen “ 2, is more interesting and involved due to the high complexity of the entries ofthe corresponding matrices. For these two examples we are able to prove both parts ofConjecture 2.30.

For the interested reader, the irregular filters Qirr for n “ 2, 3, 4, 5 and for severalvalues of hr ą 0 are available in [59].

Case n “ 1: linear B-spline scheme

In the regular case, i.e. h` “ hr “ 1, the linear B-spline scheme is defined by the mask

rppkq : k “ ´1, 0, 1s “

„

1

21

1

2

T

.

By Proposition 2.27, with rdpkq : k “ ´1, 0s ““

1 1‰

, we get

rq1pkq : k “ ´1, 0, 1s “1?

2

“

1 0 ´1‰T,

rq2pkq : k “ ´1, 0, 1s “1

2

“

´1 2 ´1‰T.

In the semi-regular case, the subdivision matrix P does not depend on h and is the 2-slanted matrix with columns determined by p. The corresponding basic limit functionsare the ones defined in (1.21), Example 1.20. Thus, the entries of D in (2.6) are welldefined for every h ą 0 and, the first moments of the scaling functions satisfy

mj,0 “ D1{2 1 “

„

. . . 1 1

c

1` h

2

?h?h . . .

T

“ cj,0.

72


The Algorithm 1 computes Sirr “ 1, thus, by (2.30), part piiq of Conjecture 2.30 is true.Moreover, by (2.39), we get

Rirr “

»

—

—

—

—

—

–

2h` 1

4ph` 1q´

?2

4?h` 1

´

?h

4ph` 1q

´

?2

4?h` 1

1

2´

?2h

4?h` 1

´

?h

4ph` 1q´

?2h

4?h` 1

h` 2

4ph` 1q

fi

ffi

ffi

ffi

ffi

ffi

fl

“ QirrQTirr,

with

Qirr “

»

—

—

—

—

—

–

´1

2?h` 1

?2h

2?h` 1

?2

20

´

?h

2?h` 1

´

?2

2?h` 1

fi

ffi

ffi

ffi

ffi

ffi

fl

.

Therefore, part piq of Conjecture 2.30 is also true.

Case n “ 2: Dubuc-Deslauriers 4-point scheme

For n “ 2, which is also a special case of the family of schemes constructed in [1, 2], dueto Definition 2.23, we obtain the regular columns of P as shifts of the regular mask

rppkq : k “ ´3, . . . , 3s “1

16

“

´1 0 9 16 9 0 ´1‰T

and the five irregular columns of P are given by

rPpm, kqs´7ďmď7kPIirr

“

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

´1{160

9{16 ´1{161 0

9{16 9{16 ´1{160 1 0

´2h` 1

16ph` 2q

3p2h` 1q

8ph` 1q

3p2h` 1q

16h´

3

8hph` 1qph` 2q0 1 0

´3h3

8p2h` 1qph` 1q

3ph` 2q

16

3ph` 2q

8ph` 1q´

h` 2

16p2h` 1q0 1 0

´1{16 9{16 9{160 1

´1{16 9{160

´1{16

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

Proposition 2.27 with

rdpkq : k “ ´3, . . . , 0s “1

4

“

1`?

3 3`?

3 3´?

3 1´?

3‰T,

73


yields

rq1pkq : k “ ´3, . . . , 3s “

?2

16

“?3´ 2 0 6´

?3 0 ´6´

?3 0

?3` 2

‰T,

rq2pkq : k “ ´3, . . . , 3s “1

16

“

1 0 ´9 16 ´9 0 1‰T.

Applying Proposition 1.37 in the regular part of the mesh, we obtain m0,0pkq “ 1,

k ă ´2, and m0,0pkq “?h, k ą 2, and, by Proposition 1.40, we get

m0,0p´2q “

d

1

120

ˆ

h´3

2

˙2

`479

480, m0,0p´1q “

c

p7´ 2hqph` 2q

15,

m0,0p0q “

c

ph` 1q3

8h, m0,0p1q “

c

p7h´ 2qp2h` 1q

15h

and m0,0p2q “

d

122

120h

ˆ

h´3

244

˙2

`479

58560h.

The expressions for m0,0p´1q and m0,0p1q imply that D in (2.6) is positive definite if

and only if h P

ˆ

2

7,7

2

˙

. Thus, the frame construction in Section 2.2 is not valid for

other h. Moreover, for the second moment we have

m0,1pkq “ k, k ă ´2, and m0,1pkq “ k?h, k ą 2,

and in the irregular part

rm0,1pkqsk“´2,...,2 “

diagprm0,0pkqsk“´2,...,2q´1

„

h3 ´ 3h2 ` 7h´ 1205

600; ´

ph` 2qp4h2 ´ 14h` 35q

75; . . .

. . .ph` 1qph´ 1qp31h2 ` 40h` 31q

600h;p2h` 1qp35h2 ´ 14h` 4q

75h;

1205h3 ´ 7h2 ` 3h´ 1

600h

.

Next, we construct Sirr to check the validity of Conjecture 2.30. The entries of Sirrdepend in an intricate way on the parameter h, thus, we work with rSirr instead, where,for

α “5ph` 1q

2, β “

37ph2 ´ 1q

12, γ “

5ph` 1q3p431h2 ` 938h` 431q

288,

74


we have

Sirr “1

αγdiagprm0,0pkqsk“´2,...,2q rSirr diagprm0,0pkqsk“´2,...,2q.

with

rSirr “ pαβ2` γq rp11T qpm, kqs´2ďm,kď2 ` α3

rpt0tT0 qpm, kqs´2ďm,kď2 ´ α2β

“`

1tT0 ` t01T˘

pm, kq‰

´2ďm,kď2

“25ph` 1q3

48

»

—

—

—

—

–

60h2` 88h` 32 60h2

` 51h` 9 60h2` 14h´ 14 23h2

´ 9h´ 14 ´14h2´ 32h´ 14

60h2` 51h` 9 60h2

` 14h` 16 60h2´ 23h` 23 23h2

´ 16h` 23 ´14h2´ 9h` 23

60h2` 14h´ 14 60h2

´ 23h` 23 60h2´ 60h` 60 23h2

´ 23h` 60 ´14h2` 14h` 60

23h2´ 9h´ 14 23h2

´ 16h` 23 23h2´ 23h` 60 16h2

` 14h` 60 9h2` 51h` 60

´14h2´ 32h´ 14 ´14h2

´ 9h` 23 ´14h2` 14h` 60 9h2

` 51h` 60 32h2` 88h` 60

fi

ffi

ffi

ffi

ffi

fl

.

Note that part piiq of Conjecture 2.30 is equivalent to the system

$

’

’

&

’

’

%

rSirr diagprm0,0pkqsk“´2,...,2q rm0,0pkqsk“´2,...,2 “ α γ r1pkqsk“´2,...,2

rSirr diagprm0,0pkqsk“´2,...,2q rm0,1pkqsk“´2,...,2 “ α γ rt0pkqsk“´2,...,2

of polynomial equations, whose validity we checked with the help of MATLAB symbolic

tool. Due to α, γ ą 0 for h P

ˆ

2

7,7

2

˙

, part piq of Conjecture 2.30 is equivalent to

checking thatrRirr “ α γ h diagpm0,0q

´1 Rirr diagpm0,0q´1

is positive semi-definite. The renormalization leads to rRirr with polynomial entries andallows for symbolic manipulations. Indeed, this way, the generalized Sylvester criterion,

confirms that rRirr is positive semi-definite for h P

ˆ

2

7,7

2

˙

. In Figure 2.7 one can see

the framelets corresponding to a possible factorization of R with h “ 2.

Remark 2.33. The value 2{7 « 0.2857 resembles the corresponding critical value in[33, 34] computed for the irregular knot insertion for the 4-point scheme. Below thiscritical value the scheme loses smoothness. This fact makes the restriction on the rangeof the stepsize h less surprising in this case. 3

Case n ą 2

Unfortunately for n “ 3 it is already too difficult to compute the moments with respectto the mesh parameter h, as for n “ 1, 2. However we approximated, with accuracy 10´6,the critical value hcrit which defines the interval ph´1

crit, hcritq available for h to constructa wavelet tight frame from the Dubuc-Deslauriers 2n-point scheme.

75


n hcrit3 2.6224824367686184 2.3590701190361015 2.2346404908393826 2.1640648862839007 2.1193821998946988 2.089127660655424

From this table, we guess that hcrit goes to 2 for n going to 8, which means thatseeking for more vanishing moments this way requires to pay a toll on the flexibility ofthe initial mesh.

Different n P t1, . . . , 8u and h P r1, hcrit ´ 106q have been considered. In the following

table are listed the minimum eigenvalues of Rirr for some choices of n and h, computedwith the command min(eig(double(¨))) of MATLAB.

(n,h) 4{3 5{3 23 ´2.6741e´ 16 ´2.0526e´ 16 ´2.0990e´ 164 ´3.3300e´ 16 ´3.8006e´ 16 ´3.1009e´ 165 ´8.4996e´ 13 ´8.4977e´ 13 ´8.4999e´ 136 ´5.3231e´ 13 ´5.3221e´ 13 ´5.3187e´ 137 ´2.0486e´ 12 ´2.0435e´ 12 ´2.0390e´ 128 ´2.3581e´ 12 ´2.3571e´ 12 ´2.3566e´ 12

For n “ 3, we still have the explicit algebraic expression of the Daubechies symboland, thus, of the regular columns of Qj. Moreover, for fixed h ą 0, we can computealgebraically the matrix Sirr, which satisfies (2.24), and, thus, we have the precise explicitexpression for Rirr, which can be proved to be positive semi-definite by the Silvestercriterion. For n “ 4, we lose the algebraic expression of the Daubechies symbol, thus,the regular columns of Qj are just approximations of the ones given by Proposition2.27. Thus, we can only have an approximate expression of Rirr, even if Sirr can bestill computed exactly. For n ě 5, also the computation of Sirr is approximate becausethe symbolic expressions involved are too long and complicated to be managed by astandard computer. These are the reasons why there is a drop in precision in the abovetable for n ě 5. After a proper thresholding, the Qirr obtained from Rirr by singularvalue decomposition has proved to be performing good for applications in Chapter 3.

76


Qirr “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

0.0000 0.0000 ´0.0000 0.0311 0.0009 0.0481 ´0.0048 ´0.0037´0.0000 ´0.0000 ´0.0000 0.0000 0.0000 0.0000 0.0000 ´0.0000´0.0000 ´0.0000 0.0000 ´0.2085 0.0029 ´0.5363 0.0452 0.0005

0.7656 0.2646 ´0.1091 0.3036 ´0.0790 0.0978 0.0404 0.0008´0.0000 ´0.0000 0.0000 ´0.8142 ´0.0832 0.2487 0.0301 0.0007´0.4246 0.2412 ´0.0333 0.2374 ´0.0390 0.0757 0.0231 0.0007´0.2950 0.1101 0.0041 0.1827 ´0.0170 0.0578 0.0130 0.0005´0.3055 0.0064 0.0552 0.2222 0.0012 0.0696 0.0076 0.0007´0.0647 ´0.3271 0.1672 0.1482 0.0578 0.0446 ´0.0161 0.0005

0.2038 ´0.6632 0.2752 0.0547 0.1147 0.0134 ´0.0406 0.0003´0.0000 0.0000 ´0.7972 ´0.0663 0.2687 ´0.0219 ´0.0766 0.0001

0.0863 0.5593 0.4887 ´0.1329 0.2274 ´0.0491 ´0.0893 ´0.0001´0.0000 0.0000 ´0.0765 ´0.0034 ´0.7045 ´0.0610 ´0.0364 ´0.0003

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000´0.0000 0.0000 ´0.0109 ´0.0015 0.1847 0.0201 0.1492 ´0.0008

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

Figure 2.7: Irregular framelets for the Dubuc-Deslauriers 4-point scheme with h “ 2corresponding, from left to right, to the columns of a possible factorizationof Rirr “ Qirr QT

irr.

77

3 Regularity Analysis via WaveletTight Frames

This last chapter is devoted to the results in [7]. The goal is to develop theoretic tools(based on the wavelet tight frames constructed in Chapter 2) for the analysis of thesmoothness of semi-regular subdivision schemes.

Since their appearance, wavelet systems became of crucial importance in several ap-plications. One of them, which is of special interest to us, is the wavelet based charac-terization of the Besov spaces Brp,qpRq, 1 ď p, q ď 8, r ą 0, and, in particular, of theHolder-Zygmund spaces, for p “ q “ 8.

Theorem 3.1 ([50], Section 6.10). Let s P p0,8q and 1 ď p, q ď 8. Assume

φk “ φ0p¨ ´ kq : k P Z(

Y

ψj,k “ 2pj´1q{2ψ1,0p2j´1¨ ´kq : k P Z, j P N

(

Ă CspRq

is a compactly supported orthogonal wavelet system with v vanishing moments. Then,for r P p0,minps, vqq,

Brp,qpRq “

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ

bj,kψj,k : takukPZ P `ppZq,!

2jpr`12´ 1pq }tbj,kukPZ}`p

)

jPNP `qpZq

+

.

Due to the perfect reconstruction property (2.2), Theorem 3.1 basically asserts thatif we have the information about the decay of the coefficients tak “ xf, φky : k P Zuand tbj,k “ xf, ψj,ky : j P N, k P Zu of a function f P L2

pRq, we can determine thesmoothness of f . To do that however one must first compute those inner products. In theregular case we are able to do that by Proposition 1.42. However, orthogonal waveletsystems are not suitable for the semi-regular setting and we can not use Proposition1.44 in this case. Proposition 1.44 is applicable to wavelet tight frames constructed inChapter 2. A question then arises naturally: can we extend Theorem 3.1 to the semi-regular wavelet tight frames? The answer is yes and we prove it in what follows. Weactually do more, providing a generalization of Theorem 3.1, in the case p “ q “ 8, forthe wider class of function systems

F “

φk : k P Z(

Y

ψj,k : j P N, k P Z(

(3.1)

with the following properties

78

3 Regularity Analysis via Wavelet Tight Frames

(I) F forms a (Parseval/normalized) tight frame for L2pRq, i.e.

f “ÿ

kPZ

xf, φkyφk `ÿ

jPN

ÿ

kPZ

xf, ψj,ky ψj,k, f P L2pRq; (3.2)

(II) there exists a constant Csupp ą 0 such that

supkPZt| supppφkq|u ď Csupp,

supkPZt| supppψj,kq|u ď Csupp 2´j, j P N;

(3.3)

(III) there exists a constant CΓ ą 0 such that for every bounded interval K Ă R thesets

Γ0pKq “ t k P Z : supppφkq XK ‰ H u,

ΓjpKq “ t k P Z : supppψj,kq XK ‰ H u, j P N,

satisfy|ΓjpKq| ď CΓp2

j|K| ` 1q, j ě 0; (3.4)

(IV) F has v P N vanishing moments, i.e.

ż

Rxn ψj,kpxq dx “ 0, n P t0, . . . , v ´ 1u, j P N, k P Z, (3.5)

and there exists a sequence of points txj,k : j P N, k P Zu such that, for every0 ď r ď v, there exists a constant Cvm,r ą 0 such that

supkPZ

ż

R|x|r|ψj,kpx` xj,kq|dx ď Cvm,r 2´jpr`

12q; (3.6)

(V) F Ă CspRq, s ą 0, and for every 0 ď r ď s there exists a constant Csm,r ą 0 suchthat

supkPZt}φk}Cru ď Csm,r,

supkPZt}ψj,k}Cru ď Csm,r 2jpr`

12q, j P N.

(3.7)

Remark 3.2. Note that (3.6) is implied by conditions (II) and (V) for 0 ď r ď s. Indeed,

79


choosing xj,k to be the midpoint of supppψj,kq, j P N, k P Z, we get

ż

R|x|r |ψj,kpx` xj,kq|dx ď Csm,0

ˆ

| supppψj,kq|

2

˙r

2j{2 | supppψj,kq|

ďCr`1supp Csm,0

2r2´jpr`

12q.

We state the assumptions (II) and (V) separately to emphasize their duality, whichbecomes even more evident in the statements of Propositions 3.5 and 3.6 in Section3.1.1. Indeed, Theorems 3.1 and 3.3 require 0 ă r ă minps, vq, where the value of sor v affects only one of the inclusions in either Proposition 3.5 or in Proposition 3.6.Moreover, stating (IV) and (V) separately, we can easily generalize our results to thecase of dual frames with the analysis frame satisfying (II) and (IV) and the synthesisframe satisfying (III) and (V). 3

Properties (II)-(V) on one hand leave freedom to the tight frame to behave differentlyat different places along R, while on the other hand they still require a sort of uniformbehaviour of the frame elements with respect to the length of the supports, the densityof the elements along R levelwise, the vanishing moments and the smoothness. Theserestrictions, however, are not such a big deal, since for most wavelet tight frames usedin applications, including the ones constructed in Chapter 2, they are easily satisfied.Indeed, (II) and (III) follow from Assumption 1 (c) and (d) together with (2.7), since thecolumns of Qj have uniformly bounded support. The structure of Qj and (2.7) are alsoresponsible for (IV) and (V), once the considered system with v vanishing moments hasbeen constructed from a Cs subdivision scheme. Our main result, Theorem 3.3, reads asfollows.

Theorem 3.3. Let s ą 0 and v P N. Assume F Ă CspRq satisfies assumptions (I)-(V)with v vanishing moments. Then, for r P p0,minps, vqq,

Br8,8pRq “

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ

bj,kψj,k : takukPZ P `8pZq,!

2jpr`12q }tbj,kukPZ}`8

)

jPNP `8pZq

+

.

This chapter is organized as follows. In Section 3.1 we give the proof of Theorem 3.3dividing it into two parts: Theorem 3.4, in Section 3.1.1, gives the proof of Theorem3.3 in the case r P p0,8qzN and Theorem 3.9, in Section 3.1.2, provides the proof forr P N. We would like to emphasize that the results in Sections 3.1.1 and 3.1.2 are true inregular, semi-regular and even irregular cases. Theorem 3.3 implies the norm equivalencebetween Besov spaces Br8,8pRq and the sequence spaces `r8,8, r P p0,8q, see Remark3.10. The proofs in Sections 3.1.1 and 3.1.2 are reminiscent of the continuous wavelettransform techniques in [20, 50] and references therein. In Section 3.2, we illustrate ourresults with several examples. In particular, we use the wavelet tight frames constructedin Chapter 2, to approximate the Holder-Zygmund regularity of semi-regular subdivisionschemes based on B-splines, the family of Dubuc-Deslauriers subdivision schemes and

80


interpolatory radial basis functions (RBFs) based subdivision.The construction of semi-regular RBFs based schemes is a generalization of [45, 46] to the semi-regular case. Wewould like to point out that such semi-regular schemes could be used for blending curvepieces with different properties.

3.1 Characterization of Holder-Zygmund SpacesBr8,8pRq via Tight Frames

3.1.1 The Case r P p0,8qzNIn this section, in Theorem 3.4 we characterize the Holder spaces Br8,8pRq “ CrpRq XL8pRq for r P p0,8qzN in terms of the function system F in (3.1). The proof of Theo-rem 3.4 follows after Propositions 3.5 and 3.6 that stress the duality between conditions(IV) and (V). Proposition 3.5, provides the inclusion ”Ě“ under assumptions (III), (V)and r P p0,minps, 1qq. Whereas Proposition 3.6 yields the other inclusion ”Ď“ underassumptions (I), (II), (IV) and r P p0, 1q. The proof of Theorem 3.4 then extends theargument of Propositions 3.5 and 3.6 to the case r ą 1, r R N. Our results show that thecontinuous wavelet transform techniques from [20, 50] and references therein are almostdirectly applicable in the irregular setting.

Theorem 3.4. Let s P p0,8q and v P N. Assume F satisfies (I)-(V) with v vanishingmoments. Then, for r P p0,minps, vqqzN,

Br8,8pRq “

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ

bj,kψj,k : pa, bq P `r8,8 with a “ takukPZ, b “ tbj,kujPN,kPZ

+

.

We start by proving the following result.

Proposition 3.5. Let s ą 0. Assume F satisfies (III) and (V).Then, for r P p0,minps, 1qq,

Br8,8pRq Ě

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ


+

.

Proof. We consider fpxq “ f0pxq ` gpxq, x P R, where

f0pxq “ÿ

kPZ

ak φkpxq and gpxq “ÿ

jPN

ÿ

kPZ

bj,k ψj,kpxq, (3.8)

with finiteCa :“ sup

kPZt|ak|u and Cb :“ sup

jPN2jpr`

12q sup

kPZt|bj,k|u. (3.9)

Since on every open bounded interval in R the sum defining f0 is finite due to (III), we

81


have f0 P CspRq Ď CrpRq due to r ă s. Moreover, by (III) and (V), we obtain

}f0}L8 ď Ca CΓ Csm,0 ă 8. (3.10)

Analogously, since r ą 0, we have

}g}L8 ď Cb CΓ Csm,0ÿ

jPN

2´jr ă 8, (3.11)

thus, f P L8pRq. Let x, h P R. By (3.9), we get

| gpx` hq ´ gpxq | ďÿ

jPN

ÿ

kPZ

| bj,k | | ψj,kpx` hq ´ ψj,kpxq |

ď Cb |h|rÿ

jPN

2´jpr`12q

|h|r

ÿ

kPZ

| ψj,kpx` hq ´ ψj,kpxq | .

Since there exists J P Z such that

2´J ă |h| ď 2´J`1 ,

we have

| gpx` hq ´ gpxq | ď Cb |h|rÿ

jPN

2pJ´jqr´j{2ÿ

kPZ

| ψj,kpx` hq ´ ψj,kpxq |

“ Cb |h|rp A ` B q ,

where

A “

J´1ÿ

j“1

2pJ´jqr´j{2ÿ

kPZ

| ψj,kpx` hq ´ ψj,kpxq | and

B “

8ÿ

j“J

2pJ´jqr´j{2ÿ

kPZ

| ψj,kpx` hq ´ ψj,kpxq | .

(3.12)

If J ď 1, A “ 0. Otherwise, for every ε ą 0 with r ă r ` ε ă minps, 1q, due to (V) wehave

| ψj,kpx` hq ´ ψj,kpxq | ď Csm,r`ε 2jpr`ε`12q |h|r`ε ď Csm,r`ε 2pj´Jqpr`εq 2j{2 2r`ε ,

and, by (III), the sum in A over k has at most

|Γjpx` hq| ` |Γjpxq| ď 2 CΓ

82


non-zero elements. Thus,

A ď 2r`ε`1 CΓ Csm,r`ε

J´1ÿ

j“1

p2´εqpJ´jq ď 4 CΓ Csm,r`ε

J´1ÿ

j“1

p2´εqj. (3.13)

Therefore, since ε ą 0, A is bounded. To conclude the proof, we observe that, by (V),

B ď 2 CΓ Csm,0

8ÿ

j“J

p2´rqj´J “ 2 CΓ Csm,01

1´ 2´r.

Thus |gpx`hq´ gpxq|{|h|r is uniformly bounded in x and h, which leads to g P Br8,8pRqand f P Br8,8pRq with }f}Br8,8 ď C }pa, bq}`r8,8 for some constant C ą 0.

Next, we give a proof of Proposition 3.6.

Proposition 3.6. Assume F Ă C0pRq with uniformly bounded tφk : k P Zu satisfies

(I), (II) and (IV) with 1 vanishing moment. Then, for r P p0, 1q,

Br8,8pRq Ď

#

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ


+

.

Proof. Consider f P Br8,8pRq X L2pRq. We choose a representative of f in (3.2) with

coefficients ak “ xf, φky and bj,k “ xf, ψj,ky. On one hand, due to (II) and the uniformboundedness of Φ, there exists Cφ ą 0 such that

|ak| ď Cφ }f}8, k P Z. (3.14)

On the other hand, with xj,k as in (IV), we can exploit the vanishing moment of thetight frame and the regularity of f to get

|bj,k| “

ˇ

ˇ

ˇ

ˇ

ż

Rfpxq ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ż

Rp fpxq ´ fpxj,kq q ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

ď }f}Br8,8

ż

R|x´ xj,k|

r|ψj,kpxq| dx

“ }f}Br8,8

ż

R|x|r |ψj,kpx` xj,kq| dx ď 2´jpr`

12q Cvm,r }f}Br8,8

(3.15)

For a general f P Br8,8pRq the claim follows by a density argument. Thus, there existsa constant C ą 0 such that }f}Br8,8 ě C}pa, bq}`r8,8 .

Remark 3.7. In Proposition 3.6, there is no need for the tight frame to be more thancontinuous: only the vanishing moment matters. The same phenomenon happens for theinclusion Ď in Theorem 3.4 - the number of vanishing moments being the key ingredientfor its proof. On the other hand, the regularity of the wavelet tight frame F plays the

83


key role both in Proposition 3.5 and in the proof of the inclusion Ě in Theorem 3.4.This explains the duality between assumptions (IV) and (V). 3

We are now ready to complete the proof of Theorem 3.4.

Proof of Theorem 3.4. For the case r P p0, 1q see Propositions 3.5 and 3.6. Let r “ n`α,with n P N and α P p0, 1q.

1st step, proof of “Ě”: similarly to Proposition 3.5, we define constants Ca and Cb asin (3.9) and make use of the estimates in (3.10) and (3.11) to conclude that f P L8pRq.The next step is to show the existence of the n-th derivative gpnq of g in (3.8). Thisfollows by uniform convergence since, for every x P R and 0 ď ` ď n ă r, by (V), wehave

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

jPN

ÿ

kPZ

bj,k ψp`qj,kpxq

ˇ

ˇ

ˇ

ˇ

ˇ

ďÿ

jPN

ÿ

kPZ

|bj,k| |ψp`qj,kpxq| ď Cb CΓ Csm,`

ÿ

jPN

2´jpr´`q ă 8,

The same argument as in Proposition 3.5 leads to gpnq P CαpRq and, thus, f P CrpRq.2nd step, proof of “Ď” resembles [43]: similarly to Proposition 3.6, we consider f P

Br8,8pRqXL2pRq and the uniform bound for |xf, φky| is obtained as in (3.14). Exploiting

the first n vanishing moments of the tight frame we have

|xf, ψj,ky| “

ˇ

ˇ

ˇ

ˇ

ż

Rfpxq ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ż

R

˜

fpxq ´ fpxj,kq ´n´1ÿ

`“1

f p`qpxj,kq

`!px´ xj,kq

`

¸

ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

ˇ

,

where the xj,k are as in (IV). Using the property of the Taylor expansion of f centeredin xj,k with the Lagrange remainder term, we have that, for every x P R, there exists ameasurable ξpxq P R, with |ξpxq ´ xj,k| ď |x´ xj,k|, such that

|xf, ψj,ky| ď

ˇ

ˇ

ˇ

ˇ

ż

R

f pnqpξpxqq

n!px´ xj,kq

n ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

.

Now we can exploit n ` 1 vanishing moments, the Holder regularity α of f pnq and (IV)

84


to get

|xf, ψj,ky| ď1

n!

ˇ

ˇ

ˇ

ˇ

ż

R

`

f pnqpξpxqq ´ f pnqpxj,kq˘

px´ xj,kqn ψj,kpxq dx

ˇ

ˇ

ˇ

ˇ

ď}f pnq}Bα8,8

n!

ż

R|ξpxq ´ xj,k|

α|x´ xj,k|

n|ψj,kpxq| dx

ď}f pnq}Bα8,8

n!

ż

R|x´ xj,k|

r|ψj,kpxq| dx

ď}f pnq}Bα8,8

n!

ż

R|y|r |ψj,kpy ` xj,kq| dy ď 2´jpr`

12qCvm,r }f

pnq}Bα8,8

n!.

(3.16)

Thus, the claim follows.

Remark 3.8. If s P N and v ě s in Theorem 3.4, then the wavelet tight frame F does notneed to belong to CspRq. It suffices to have F Ă Cs´1

pRq with the ps´ 1q-st derivativesof its elements being Lipschitz-continuous. 3

3.1.2 The Case r P NIt is well known [50] that the Holder spaces with integer Holder exponents cannot becharacterized via either a wavelet or a wavelet tight frame system F . Indeed, if r “ 1,the estimate (3.13) does not follow from (3.9). Thus, similarly to Theorem 3.1, thenatural spaces in this context are the Holder-Zygmund spaces Br8,8pRq for r P N. Thissection is devoted to the proof of the wavelet tight frame characterization of such spaces,see Theorem 3.9. The results of Theorems 3.4 and 3.9 yield Theorem 3.3.

Theorem 3.9. Let s ą 0 and v P N. Assume F Ă CspRq satisfies (I)-(V) with vvanishing moments. Then, for 0 ă r ă minps, vq, r P N,

Br8,8pRq “!

ÿ

kPZ

akφk `ÿ

jPN

ÿ

kPZ


)

.

The significant case is when r “ 1, since for all other integers one usually arguessimilarly to the proof of Theorem 3.4. In the case r “ 1, for the inclusion Ě we use theargument similar to the one in Proposition 3.5. On the other hand, for the inclusion Ď,we cannot exploit the vanishing moments as done in (3.15). To circumvent this problem,inspired by [58], we consider an auxiliary orthogonal wavelet system which satisfies theassumptions of Theorem 3.1. This way we get a convenient expansion for f P Br8,8pRqand make use of the wavelet characterization of Bα8,8pRq for α P pr, sqzN in Theorem3.4.

Proof. We only prove the claim for r “ 1 ă minps, vq. In this case Br8,8pRq “ ΛpRq XL8pRq. The general case follows using an argument similar to the one in the proof ofTheorem 3.4.

85


1st step, proof of “Ě”: similarly to Proposition 3.5, we define constants Ca and Cb asin (3.9) and make use of the estimates in (3.10) and (3.11) to conclude that f P L8pRq.Let x P R. It suffices to consider h ą 0. Then, for r “ 1, we obtain

| gpx` hq ´ 2 gpxq ` gpx´ hq | ď

ďÿ

jPN

ÿ

kPZ

| bj,k | | ψj,kpx` hq ´ 2 ψj,kpxq ` ψj,kpx´ hq |

ď Cb hÿ

jPN

2´j32

h

ÿ

kPZ

| ψj,kpx` hq ´ 2 ψj,kpxq ` ψj,kpx´ hq | .

Since there exists J P Z such that

2´J ă h ď 2´J`1 ,

we have

| gpx` hq ´ 2 gpxq ` gpx´ hq | ď

ď Cbhÿ

jPN

2J´j32

ÿ

kPZ


To estimate | gpx` hq ´ 2 gpxq ` gpx´ hq |, we consider

A “

J´1ÿ

j“1

2J´j32

ÿ

kPZ

| ψj,kpx` hq ´ 2 ψj,kpxq ` ψj,kpx´ hq |

and

B “

8ÿ

j“J

2J´j32

ÿ

kPZ


If J ď 1, A “ 0. Otherwise, since the tight frame F belongs to CspRq, s ą 1, weuse the mean value theorem twice for every framelet and find ξj,kpxq P rx, x ` hs andηj,kpxq P rx´ h, xs such that

| ψj,kpx` hq ´ 2 ψj,kpxq ` ψj,kpx´ hq | “

“ | ψj,kpx` hq ´ ψj,kpxq ´ p ψj,kpxq ´ ψj,kpx´ hq q |

“ hˇ

ˇ ψ1j,kpξj,kpxqq ´ ψ1j,kpηj,kpxqqˇ

ˇ .

86


Now, for ε ą 0 with r “ 1 ă 1` ε ă s, using (V) we get

| ψj,kpx` hq ´ 2 ψj,kpxq ` ψj,kpx´ hq | “

“ Csm,1`ε 2jpε`32q h | ξj,kpxq ´ ηj,kpxq |

ε

ď Csm,1`ε 2jpε`32q`ε h1`ε

ď Csm,1`ε 2pj´Jqp1`εq`j2`1`2ε.

Moreover, the sum in A over k has at most

|Γjpx` hq| ` |Γjpx´ hq| ` |Γjpxq| ď 3 CΓ

non-zero summands and, thus, we get

A ď CΓ Csm,1`ε 21`2ε 3J´1ÿ

j“1

p2´εqJ´j.

Since ε ą 0, A is bounded. To conclude the proof, we observe that

B ď CΓ Csm,0 38ÿ

j“J

2J´j “ 6 CΓ Csm,0.

Thus, g P B18,8pRq and, therefore, f P B1

8,8pRq with }f}B18,8

ď C}pa, bq}`18,8 for someconstant C ą 0.

2nd step, proof of “Ď”: similarly to Proposition 3.6 we only consider f P ΛpRqXL2pRq.

The uniform bound for |xf, φky| is obtained similarly to (3.14). To obtain the bound for

|xf, ψj,ky|, we let rΦ Y trΨ`u`PN Ď CspRq be an auxiliary compactly supported orthogonalwavelet system with v vanishing moments (e.g. Daubechies 2n-tap wavelets [20] with

large enough n P N). rΦ Y trΨ`u`PN satisfies the assumptions of Theorem 3.1 and fulfills

(I)-(V), with appropriate rΓj and rCsupp ą 0, rCΓ ą 0, rCvm,r ą 0 and rCsm,r ą 0. Then,

from (3.2), we have fpxq “ rf0pxq ` rgpxq, x P R, where

rf0pxq “ÿ

mPZ

xf, rφmy rφmpxq and rgpxq “ÿ

`PN

ÿ

mPZ

xf, rψ`,my rψ`,mpxq.

Thus, for every j P N and k P Z, we get

|xf, ψj,ky| ďˇ

ˇ

ˇx rf0, ψj,k y

ˇ

ˇ

ˇ`

ÿ

`PN

ÿ

mPZ

|xf, rψ`,my| |x rψ`,m, ψj,ky| .

Let α P p1, sqzN. Since both tight frames belong to CspRq Ď CαpRq, the function rf0,which is locally the finite sum of Cα-functions, belongs to CαpRq, and, by Theorem 3.4,

87


there exists C1 ą 0, such that

supkPZ|x rf0, ψj,ky| ď C1 2´jpα`

12q ď C1 2´j

32 , j P N.

Moreover, by Theorem 3.1 and due to f P ΛpRq, there exists C2 ą 0 such that

supmPZ

|xf, rψ`,my| ď C2 2´`32 , ` P N.

Thus,

|xf, ψj,ky| ď C1 2´j32 ` C2

ÿ

`PN

2´`32

ÿ

mPZ

|x rψ`,m, ψj,ky|

“ C1 2´j32 ` C2

˜

j´1ÿ

`“1

2´`32

ÿ

mPZ

|x rψ`,m, ψj,ky| `8ÿ

`“j

2´`32

ÿ

mPZ

|x rψ`,m, ψj,ky|

¸

“ C1 2´j32 ` C2 p A ` B q .

(3.17)

The sums in (3.17) over m have at most

|rΓ`psupppψj,kqq| ď rCΓ p 2` |supppψj,kq| ` 1 q ď rCΓ p Csupp 2`´j ` 1 q

non-zero summands. When ` ă j, by assumption (V) for rψ`,m and (3.16) with f “ rψ`,m,due to Theorem 3.4, we have

|x rψ`,m, ψj,ky| ď Cvm,α 2´jpα`12q} rψ`,m}Cα ď Cvm,α rCsm,α 2p`´jqpα`

12q,

uniformly in m and k. Thus, substituting `1 “ j ´ `, we obtain

A ď Cvm,α rCΓrCsm,α

j´1ÿ

`“1

2´`32 2p`´jqpα`

12q p Csupp 2`´j ` 1 q

“ Cvm,α rCΓrCsm,α 2´j

32

j´1ÿ

`1“1

2´`1pα´1q

p Csupp 2´`1

` 1 q

ď C3 2´j32 ,

(3.18)

for some C3 ą 0, due to the fact that α ą 1.On the other hand, when ` ě j, using (II) and (V), we get

|x rψ`,m, ψj,ky| ď Csm,0 rCsm,0 2pj``q{2 min´

Csupp2´j, rCsupp2

´`¯

“ C4 2pj´`q{2,

88


uniformly in m and k. Thus, after the substitution `1 “ `´ j, we obtain

B ď C4rCΓ

8ÿ

`“j

2´`32 p Csupp 2`´j ` 1 q 2pj´`q{2

“ C4rCΓ 2´j

32

8ÿ

m“0

2´2mp Csupp 2m ` 1 q ď C5 2´j

32

(3.19)

for some constant C5 ą 0. Combining (3.17), (3.18) and (3.19) we finally get

supkPZ|xf, ψj,ky| ď pC1 ` C2C3 ` C2C5q 2´j

32 , j P N.

Thus, the claim follows, i.e., there exists a constant C ą 0 such that }f}B18,8

ě

C}pa, bq}`18,8 .

Remark 3.10. The norm equivalence between the Besov norm } ¨ }Br8,8 and } ¨ }`r8,8 ,r P p0,8q, is a consequence of Theorem 3.3 and the Open Mapping Theorem. 3

3.2 Approximation of the Optimal Holder-ZygmundExponent

In this section, we show how to apply Theorem 3.3 for estimating the Holder-Zygmundregularity of a semi-regular subdivision scheme from the decay of the frame coefficientsof its basic limit functions with respect to a given tight frame F satisfying (I)-(V) forsome s ą 0 and v P N. In Section 3.2.1, in a general irregular setting, we discusshow to obtain such regularity estimates using the result of Theorem 3.3. In Section3.2.2, we describe how to compute the frame coefficients in the semi-regular case usingProposition 1.44. Lastly, in Section 3.2.3, we illustrate our results with examples ofsemi-regular schemes such as B-spline, Dubuc-Deslauriers subdivision and interpolatoryschemes based on radial basis functions. The latter example in the regular settingreduces to the construction in [45].

3.2.1 Two Methods for the Estimation of the OptimalHolder-Zygmund Exponent

Definition 3.11. Let f P L8pRq. We call optimal Holder-Zygmund (smoothness) expo-nent of f the real number

rpfq “ supt r ą 0 : f P Br8,8pRq u.

89


Assume that rpfq P p0,minps, vqq and that we are given

γj “ supkPZ

|xf, ψj,ky|, j P N.

By Theorem 3.3, for every ε ą 0, there exists a constant Cε ą 0 such that, for everyj P N,

γj ď Cε 2´jprpfq´ε`12q, i.e. j

ˆ

rpfq ´ ε`1

2

˙

´ log2pCεq ď ´ log2pγjq. (3.20)

From (3.20) we infer that searching for rpfq is equivalent to searching for the largestslope of a line lying under the set of points tpj,´ log2pγjqqujPN. With this interpretationin mind, the natural approach (see e.g. [20]) to approximate rpfq is to compute the real-valued sequence trnpfqunPN, where rnpfq ´ 1{2 is the slope of the regression line for thepoints tpj,´ log2pγjqqu

n`1j“1 . This method is robust, i.e. for larger n the contributions of

the levels j ě n become less significant, thus, the difference between rnpfq and rn`1pfq issmall and we are able to estimate the overall distribution of tpj,´ log2pγjqqujPN. However,examples in Subsection 3.2.3 illustrate that the convergence of trnpfqunPN towards rpfqis very slow. One of the main reasons for such a behavior is the value of the unknownCε, which can be significant, e.g when f R Brpfq8,8pRq.

An alternative approach for estimating the Holder-Zygmund exponent is given by thefollowing Proposition.

Proposition 3.12. Let rpfq be the optimal Holder-Zygmund exponent of f P C0pRq. If

0 ă r˚pfq “ limnÑ8

log2

ˆ

γnγn`1

˙

´1

2ă minps, vq ,

then r˚pfq “ rpfq.

Proof. We first prove that r˚pfq ď rpfq and then, by contradiction, that r˚pfq “ rpfq.Let ε ą 0. We consider the series

Spr˚pfq ´ εq “8ÿ

j“1

2jpr˚pfq´ε` 1

2q γj. (3.21)

By the assumption, we obtain

limnÑ8

2pn`1qpr˚pfq´ε` 12q γn`1

2npr˚pfq´ε` 1

2q γn“ 2r

˚pfq´ε` 12 limnÑ8

γn`1

γn“ 2´ε . (3.22)

Thus, by the ratio test, the series Spr˚pfq ´ εq in (3.21) converges for every ε ą 0.Consequently, the non-negative summands of Spr˚pfq ´ εq are uniformly bounded, i.e.there exists Cε ą 0 such that

γj ď Cε2´jpr˚pfq´ε` 1

2q, j P N.

90


Therefore, by Definition 3.11 and by (3.20), r˚pfq ď rpfq.On the other hand, by (3.20), if r˚pfq ă rpfq, then there exists δ ą 0 and a constantCδ ą 0 such that

γj ď Cδ 2´jpr˚pfq`δ` 1

2q, j P N.

Therefore, similarly to (3.22), we obtain that the series Spr˚pfq ` δ{2q diverges and atthe same time is bounded from above by

Spr˚pfq ` δ{2q “8ÿ

j“1

2jpr˚pfq` δ

2` 1

2q γj ď Cδ

8ÿ

j“1

2´jδ{2 .

Thus, due to this contradiction, r˚pfq “ rpfq.

The advantage of the approach in Proposition 3.12 is that it eliminates the effect ofthe constant Cε in (3.20). Even though the existence of r˚pfq is not guaranteed and theelements of the sequence

r˚npfq “ log2

ˆ

γnγn`1

˙

´1

2, n P N,

can oscillate wildly, our numerical experiments in Subsection 3.2.3 provide exampleswhich illustrate the cases when tr˚npfqunPN converges to rpfq rapidly. The convergencein these examples is much faster than that of the linear regression method.

Remark 3.13. The series in (3.21) with ε “ 0 becomes

Sprq “

›

›

›

›

!

2jpr`12q

›

› t x f, ψj,k y ukPZ›

›

`8

)

jPN

›

›

›

›

`1.

This norm appears in the characterization of Br8,1pRq in Theorem 3.1 and correspondsto pa, bq P `r8,1, r P p0,8q. Even if the case p “ 8 and q “ 1 is not covered by Theorem3.3, this observation is consistent with Brp,q1pRq Ď Brp,q2pRq for q1 ď q2. 3

3.2.2 Computation of Frame Coefficients

Conditions (I)-(V) do not require the semi-regularity of the mesh and all the aboveresults hold even in the irregular case. To use the presented results in practice, however,we need an efficient method for computing the frame coefficients

tak “ xf, φkyukPN and tbj,k “ xf, ψj,kyujPN,kPZ.

If F as in (2.7) is a wavelet tight frame obtained from a convergent subdivision schemewith subdivision matrix P and basic limit functions tϕkukPZ as in Chapter 2 and thefunction f we want to analyse is a limit function of another convergent semi-regularsubdivision scheme with subdivision matrix Z and basic limit functions tζkukPZ, we cando it in a rather simple way exploiting Proposition 1.44 and the following result.

91


Proposition 3.14. For every j P N, i, k P Z, we have ă ζk, ψj,m ą “ Cjpk,mq, where

Cj “ 2´j{2 pZjqT G D´1{2 Qj,

with the cross-Gramian G “ rxζk, ϕmysk,mPZ, D as in (2.6) and Qj are the matricesdefining the wavelet tight frame in (2.7).

Proof. Applying (2.7), the substitution y “ 2jx, (1.17) and (2.5), we get

ż

RrζkpxqskPZ Ψjpxq

T dx “ 2j{2ż

RrζkpxqskPZ Φ0p2

jxqT dx Qj

“ 2´j{2ż

Rrζkp2

´jyqskPZ Φ0pyqT dy Qj

“ 2´j{2 pZjqT

ż

RrζkpyqskPZ Φ0pyq

T dy Qj

“ 2´j{2 pZjqT

ż

RrζkpyqskPZ rϕmpyqs

TmPZ dy D´1{2 Qj

“ 2´j{2 pZjqT G D´1{2 Qj.

In practice then, given a wavelet frame, we only need to compute the cross-Gramianmatrix G and then we can obtain the frame coefficients of tζkukPZ just by matrix mul-tiplication. The only thing one has to be careful about is to cut properly the matricesat each step, since the products in Proposition 3.14 are between bi-infinite matrices.

3.2.3 Numerical Estimates

For simplicity of presentation, in this subsection we choose h` “ 1 and hr “ 2 forthe initial semi-regular mesh t0 (1.3). The wavelet tight frames used for our numericalexperiments are the ones constructed in Section 2.2 from the Dubuc-Deslauriers 2n-point subdivision schemes. We present an application of the methods in Section 3.2.1to four cases: the quadratic B-spline scheme, the Dubuc-Deslauriers 4-point scheme andthe semi-regular version of two interpolatory schemes based on radial basis functions.The optimal Holder-Zygmund exponents of semi-regular B-splines schemes and Dubuc-Deslauriers 4-point scheme are known. These examples are used as a benchmark to testour theoretical results.

Example 3.15. The quadratic B-spline scheme generates basic limit functions whichare piecewise polynomials of degree two, supported between four consecutive knots oft0. The corresponding subdivision matrix Z is constructed to satisfy these conditions.There are krpZq ´ k`pZq ´ 1 “ 2 irregular functions whose supports contain the point

92


tp0q “ 0 and it is well known that these functions are C2´εpRq, ε ą 0, thus their optimal

exponent is equal to 2.In Figure 3.1 we give the estimates of the optimal Holder-Zygmund exponents by

both the linear regression method and by the method in Proposition 3.12. For theanalysis we used the semi-regular tight wavelet frame constructed from the limits of thesemi-regular Dubuc-Deslauriers 6-point subdivision scheme. This toy example alreadyillustrates that the method proposed in Proposition 3.12 reaches the optimal exponentin few steps, while the linear regression method converges much slower. 4

Example 3.16. The scheme considered here is the one obtained in Definition 2.23 withn “ 2. In this case, there are 5 irregular basic limit functions depicted in Figure 3.2.Due to results in [21], it is well known that the optimal exponent of all these irregularfunctions is equal to 2. Again, the method in Proposition 3.12 remarkably outperformsthe linear regression method.

4

Radial basis functions based interpolatory schemes

Using techniques similar to the ones in Definition 2.23, we extend the subdivision schemes[45, 46] based on radial basis functions to the semi-regular setting. Let L P N. We requirethat the subdivision matrix Z satisfies Zp2i, kq “ δi,k for i, k P Z. To determine the otherentries of the 2-slanted matrix Z whose columns are centered at Zp2k, kq, k P Z and havesupport length at most 4L ´ 1, we proceed as follows. We first choose a radial basisfunction gpxq “ gp|x|q, x P R, which is conditionally positive definite of order η P N, i.e.,for every set of pairwise distinct points txiu

Ni“1 Ă R and coefficients tciu

Ni“1 Ă R, N P N,

there exists a polynomial π of degree at most η ´ 1 such that

Nÿ

i“1

ci πpxiq “ 0

and the function g satisfies

Nÿ

i“1

Nÿ

k“1

ci ck gpxi ´ xkq ě 0.

The next step is to choose the order m P tη, . . . , 2Lu of polynomial reproduction and,for every set of 2L consecutive points t0pk ´ L` 1q, . . . , t0pk ` Lq, k P Z, of the mesht0 in (1.3), solve the linear system of equations

»

–

A B

BT 0

fi

fl

»

–

u

v

fi

fl “

»

–

r

s

fi

fl (3.23)

witht Api, jq “ gpt0pk ´ L` iq ´ t0pk ´ L` jqq ui,j“1,...,2L,

93


t Bpi, jq “ t0pk ´ L` iqj´1

ui“1,...,2L, j“1,...,m,

t rpiq “ gpxk ´ t0pk ´ L` iqq ui“1,...,2L, t spjq “ xj´1k uj“1,...,m and

xk “ p t0pkq ` t0pk ` 1q q{2.

Lastly, the vector u contains the entries of the p2k ` 1q-th row of Z associated to thecolumns k ´ L ` 1 to k ` L. For the interested reader, a MATLAB function for thegeneration of semi-regular RBFs-based interpolatory schemes is available at [61].

Remark 3.17. piq Determining the rows of Z by solving the linear systems (3.23) fork P Z guarantees the polynomial reproduction of degree at most m ´ 1. Indeed, thecondition BT u “ s forces Z to map samples over t0 of a polynomial of degree at mostm´ 1 onto sample over the finer knots t0{2 of the same polynomial.piiq If m “ 2L, the system of equations BT u “ s coincides with the one definingthe Dubuc-Deslauriers 2L-point scheme in Definition 2.23. In this case, the systemBT u “ s has a unique solution, which makes the presence of A, i.e. of the radial basisfunction g obsolete.piiiq In general, if m ă 2L, the structure of the irregular basic limit functions aroundt0p0q reflects the transition (blending) between the two (one on the left and one onthe right of t0p0q) subdivision schemes of different regularity, see Example 3.18. Thisdepends on the properties of the chosen underlying radial basis function g. For example,the blending produces no visible effect if g is homogeneous, i.e. gpλxq “ |λ|gpxq, λ P R.In this case, for λ ą 0, the linear system of equations

»

–

λI 0

0 L

fi

fl

»

–

A B

BT 0

fi

fl

»

–

I 0

0 L{λ

fi

fl

»

–

I 0

0 λL´1

fi

fl

»

–

u

v

fi

fl “

»

–

λI 0

0 L

fi

fl

»

–

r

s

fi

fl (3.24)

with the identity matrix I and L “ diagprλj´1sj“1,...,mq is equivalent to the system in

(3.23) for the mesh λt0. The structure of the linear system in (3.24) implies that u isthe same as the one determined by (3.23).pivq The argument in piiiq with L “ I shows that the subdivision matrix obtained thisway does not depend on the normalization of the radial basis function g, i.e. all functionsλg, λ ą 0, lead to the same subdivision scheme. 3

Example 3.18. We consider the radial basis function introduced by M. Buhmann in[4]

gpxq “

$

&

%

12x4 log |x| ´ 21x4` 32|x|3 ´ 12x2

` 1, if |x| ă 1,

0, otherwise,(3.25)

and choose L “ 2 and m “ 1. The resulting irregular functions ζ´2, . . . , ζ2 are shownin Figure 3.3. The structure of ζ0 illustrates the blending effect (described in Remark3.17 part piiiq) of two different subdivision schemes meeting at t0p0q. Figure 3.3 alsopresents the estimates of the optimal Holder-Zygmund exponents of ζ´2, . . . , ζ2. These

94


exponents are determined using the tight wavelet frame based on the Dubuc-Deslauriers4-point subdivision scheme (see Example 3.16). We again observe the phenomenon thatthe method in Proposition 3.12 converges faster than the linear regression. 4

Example 3.19. Another radial basis function that we consider is the polyharmonicfunction gpxq “ |x|, x P R. The corresponding irregular part of the interpolatorysubdivision matrix Z is determined for L “ 2 and m “ 3, see Figure 3.4. Note that theregular part of the subdivision matrix Z (see the first and the last columns correspondingto the regular parts of the mesh) coincides with the subdivision matrix of the regularDubuc-Deslauriers 4-point scheme. Due to the observation in Remark 3.17 part piiiq,the absence of the blending effect is due to our choice of a homogeneous function g. Wewould like to emphasise that the resulting subdivision scheme around t0p0q is not thesemi-regular Dubuc-Deslauriers 4-point scheme, compare with Figure 3.2. Indeed, thepolynomial reproduction around t0p0q is of one degree lower. We also lose regularity(the Dubuc-Deslauriers 4-point scheme is C2´ε, ε ą 0) but overall the irregular limitfunctions on Figure 3.4 have a more uniform behavior than those in Figure 3.2. Weagain observe that the method in Proposition 3.12 yields better estimates for the optimalHolder-Zygmund exponent, see tables on Figure 3.4. 4

To conclude, we tested this method for a large number of other semi-regular familiesof subdivision schemes (i.e. B-splines, Dubuc-Deslauriers and interpolatory schemesbased on (inverse) multi-quadrics, gaussians, Wendland’s functions, Wu’s functions,Buhmann’s functions, polyharmonic functions and Euclid’s hat functions [32]) obtainingsimilar results.

95


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1{43{43{4 1{41{4 3{4

5{6 1{61{3 2{3

3{4 1{41{4 3{4

3{41{4

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-2 -1 0 2 4

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n rnpζ´2q rnpζ´1q

1 -0.9004 0.83332 -0.0362 0.52353 0.6611 0.93554 1.0683 1.23085 1.3178 1.4250

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1 -0.9004 0.83332 0.8280 0.21363 2.0000 2.00014 2.0000 2.00005 2.0000 2.0000

1 2 3 4 5

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r

Figure 3.1: Semi-regular quadratic B-spline functions on the mesh t0 with h` “ 1and hr “ 2 analyzed with the semi-regular Dubuc-Deslauriers 6-point tightwavelet frame. Top row: part of the subdivision matrix Z that correspondsto the non-shift-invariant refinable functions around t0p0q and the graphs ofthese functions ζ´2 and ζ´1. Middle row: estimates of the optimal Holder-Zygmund exponents of ζ´2 and ζ´1 via linear regression (on the left) viathe method in Proposition 3.12 (on the right). Bottom row: graphs of theestimates of the Holder-Zygmund exponents.

96


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9{16 ´1{161 0

9{16 9{16 ´1{160 1 0

´1{16 9{16 9{16 ´1{160 1 0

´5{64 5{8 15{32 ´1{640 1 0

´1{5 3{4 1{2 ´1{200 1 0

´1{16 9{16 9{16 ´1{160 1 0

´1{16 9{16 9{160 1

´1{16 9{160

´1{16

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-5 -4 -3 -2 -1 0 2 4 6 8 10

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n rnpζ´2q rnpζ´1q rnpζ0q rnpζ1q rnpζ2q

1 -0.0589 -0.6276 -0.4561 -0.6674 1.47062 1.5123 0.4586 0.6302 0.8716 1.83563 1.8333 1.0375 1.1799 1.5818 2.22404 1.9052 1.3336 1.4487 1.8532 2.35235 1.9360 1.5117 1.6052 1.9587 2.35966 1.9516 1.6266 1.7032 2.0022 2.32397 1.9615 1.7052 1.7688 2.0209 2.28208 1.9683 1.7614 1.8148 2.0286 2.24369 1.9734 1.8030 1.8484 2.0310 2.210810 1.9774 1.8346 1.8736 2.0310 2.183311 1.9805 1.8592 1.8929 2.0298 2.160412 1.9830 1.8786 1.9082 2.0281 2.141313 1.9851 1.8944 1.9204 2.0262 2.125214 1.9868 1.9072 1.9303 2.0244 2.111615 1.9882 1.9179 1.9385 2.0226 2.100116 1.9895 1.9268 1.9453 2.0209 2.0902

n r˚npζ´2q r˚npζ´1q r˚npζ0q r˚npζ1q r˚npζ2q

1 -0.0589 -0.6276 -0.4561 -0.6674 1.47062 3.0835 1.5448 1.7165 2.4105 2.20073 2.0585 2.0261 2.1006 2.7262 3.00864 1.8721 1.9392 1.9742 2.2284 2.47695 1.9764 1.9883 2.0064 2.0489 2.14746 1.9792 1.9897 1.9984 2.0131 2.00137 1.9920 1.9960 2.0004 2.0096 1.99978 1.9954 1.9977 1.9999 2.0041 2.00019 1.9979 1.9989 2.0000 2.0022 2.000010 1.9989 1.9995 2.0000 2.0011 2.000011 1.9995 1.9997 2.0000 2.0005 2.000012 1.9997 1.9999 2.0000 2.0003 2.000013 1.9999 1.9999 2.0000 2.0001 2.000014 1.9999 2.0000 2.0000 2.0001 2.000015 2.0000 2.0000 2.0000 2.0000 2.000016 2.0000 2.0000 2.0000 2.0000 2.0000

5 10 15n

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Figure 3.2: Semi-regular Dubuc-Deslauriers 4-point limit functions on the mesh t0 with h` “ 1 andhr “ 2 analyzed via the semi-regular Dubuc-Deslauriers 6-point tight wavelet frame.Top row: part of the subdivision matrix Z that corresponds to the non-shift-invariantrefinable functions around t0p0q and the graphs of these functions ζ´2, . . . , ζ2. Middle row:estimates of the optimal Holder-Zygmund exponents of ζ´2, . . . , ζ2 via linear regression(on the left) via the method in Proposition 3.12 (on the right). Bottom row: graphs ofthe estimates of the Holder-Zygmund exponents.

97


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0.2500 0.2500 0.2500 0.25000 1 0

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1 2.3107 0.0955 -0.3137 1.9167 1.17512 1.3713 0.7568 0.1025 1.2398 0.73053 1.0062 0.5966 0.2212 0.9391 0.56094 0.8229 0.5853 0.2821 0.7386 0.47545 0.7185 0.5437 0.3146 0.6290 0.42696 0.6532 0.5226 0.3342 0.5552 0.39657 0.6097 0.5014 0.3466 0.5037 0.37638 0.5793 0.4860 0.3550 0.4664 0.36229 0.5571 0.4726 0.3608 0.4387 0.351910 0.5405 0.4618 0.3650 0.4175 0.344211 0.5278 0.4525 0.3681 0.4010 0.338312 0.5177 0.4448 0.3705 0.3879 0.333613 0.5097 0.4382 0.3723 0.3773 0.329914 0.5032 0.4325 0.3737 0.3686 0.326915 0.4978 0.4276 0.3749 0.3614 0.324416 0.4934 0.4234 0.3758 0.3554 0.3223


1 2.3107 0.0955 -0.3137 1.9167 1.17512 0.4319 1.4181 0.5186 0.5630 0.28593 0.4673 0.0025 0.3595 0.4631 0.31334 0.4549 0.7000 0.4071 0.2370 0.30295 0.4585 0.3168 0.3879 0.3727 0.30696 0.4573 0.4855 0.3888 0.3053 0.30537 0.4577 0.3743 0.3860 0.3059 0.30598 0.4576 0.4187 0.3846 0.3057 0.30579 0.4576 0.3832 0.3839 0.3058 0.305810 0.4576 0.3957 0.3831 0.3057 0.305711 0.4576 0.3838 0.3829 0.3058 0.305812 0.4576 0.3872 0.3825 0.3058 0.305813 0.4576 0.3832 0.3824 0.3058 0.305814 0.4576 0.3841 0.3823 0.3058 0.305815 0.4576 0.3827 0.3822 0.3058 0.305816 0.4576 0.3829 0.3822 0.3058 0.3058

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Figure 3.3: Semi-regular interpolatory subdivision scheme based on g in (3.25), L “ 2, m “ 1, onthe mesh t0 with h` “ 1 and hr “ 2 analyzed via the semi-regular Dubuc-Deslauriers4-point tight wavelet frame. Top row: part of the subdivision matrix Z that correspondsto the non-shift-invariant refinable functions around t0p0q and the graphs of these func-tions ζ´2, . . . , ζ2. Middle row: estimates of the optimal Holder-Zygmund exponents ofζ´2, . . . , ζ2 via linear regression (on the left) via the method in Proposition 3.12 (on theright). Bottom row: graphs of the estimates of the Holder-Zygmund exponents.

98


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´1{24 19{36 13{24 ´1{360 1 0

´1{9 7{12 11{18 ´1{120 1 0

´1{16 9{16 9{16 ´1{160 1 0

´1{16 9{16 9{160 1

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1 -0.0163 -0.5174 -0.3984 -0.5637 1.45252 1.8110 0.6301 0.2509 0.5646 1.53243 2.3259 0.9516 0.7115 1.0643 1.68484 2.1635 1.1421 1.0006 1.3048 1.73405 2.0346 1.2779 1.1895 1.4400 1.75526 1.9538 1.3766 1.3178 1.5230 1.76407 1.9029 1.4495 1.4081 1.5777 1.76778 1.8695 1.5043 1.4740 1.6155 1.76929 1.8466 1.5463 1.5232 1.6429 1.769610 1.8303 1.5791 1.5610 1.6633 1.769511 1.8183 1.6050 1.5906 1.6789 1.769312 1.8092 1.6260 1.6141 1.6911 1.768913 1.8022 1.6430 1.6332 1.7008 1.768514 1.7967 1.6571 1.6488 1.7087 1.768115 1.7922 1.6689 1.6618 1.7152 1.767716 1.7886 1.6788 1.6727 1.7206 1.7674


1 -0.0163 -0.5174 -0.3984 -0.5637 1.45252 3.6383 1.7777 0.9002 1.6929 1.61243 2.9182 1.3190 1.5699 1.8541 2.01354 0.9991 1.5830 1.6965 1.7672 1.77875 1.5859 1.7110 1.7445 1.7700 1.78386 1.7105 1.7467 1.7568 1.7642 1.76657 1.7468 1.7580 1.7612 1.7638 1.76498 1.7580 1.7615 1.7625 1.7633 1.76369 1.7615 1.7626 1.7630 1.7632 1.763310 1.7626 1.7630 1.7631 1.7632 1.763211 1.7630 1.7631 1.7632 1.7632 1.763212 1.7631 1.7632 1.7632 1.7632 1.763213 1.7632 1.7632 1.7632 1.7632 1.763214 1.7632 1.7632 1.7632 1.7632 1.763215 1.7632 1.7632 1.7632 1.7632 1.763216 1.7632 1.7632 1.7632 1.7632 1.7632

5 10 15n

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Figure 3.4: Semi-regular interpolatory scheme based on the polyharmonic function gpxq “ |x|, L “ 2and m “ 3 on the mesh t0 with h` “ 1, hr “ 2 analyzed with the semi-regular Dubuc-Deslauriers 6-point scheme. Top row: part of the subdivision matrix Z that correspondsto the non-shift-invariant refinable functions around t0p0q and the graphs of these func-tions ζ´2, . . . , ζ2. Middle row: estimates of the optimal Holder-Zygmund exponents ofζ´2, . . . , ζ2 via linear regression (on the left) via the method in Proposition 3.12 (on theright). Bottom row: graphs of the estimates of the Holder-Zygmund exponents.

99

Conclusions

We presented a new characterization of the Holder-Zygmund spaces via irregular familiesof tight frames, in particular the one developed from semi-regular subdivision schemes.We provided the tools for the computation of the moments and the (Cross-)Gramianmatrices involving basic limit functions of such semi-regular schemes. This opened upthe possibility for estimating the regularity of any semi-regular scheme via the decayof the frame coefficients of its basic limit functions. For the analysis we constructedwavelet tight frames associated with the Dubuc-Deslauriers family of semi-regular inter-polatory subdivision schemes. We presented their convergence analysis, to the choice ofa suitable approximation of the corresponding Gramian matrix. This UEP constructionwas developed to overcome the difficulties arising in the OEP semi-regular setting andits simplicity may also have a strength in other practical applications.

The results presented here can be generalized in a straightforward way to tensorproducts of semi-regular schemes in the bivariate case. This could be the first steptowards the analysis of the bivariate subdivision in presence of extraordinary vertices.

100

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105

Index

(cross-)Gramian matrix, 26, 45, 93regular, 31semi-regular, 32

autocorrelation function, 32

B-splinecubic, 22, 23, 42, 51linear, 11, 15, 17, 21, 73quadratic, 93, 97

basic limit functionsevaluation, 20integral, 28regular, 16renormalization, 40semi-regular, 16support, 22

Besov spaces, 8wavelet characterization, 79

Bessel sequence, 44

Cholesky factorization, 59

Daubechies wavelets, 67Dubuc-Deslauriers

4-point scheme, 74, 94, 98schemes, 64

eigenvaluesleft, 30right, 13, 14

eigenvectorsleft, 20, 30right, 13

extension principles, 41

frame, 36coefficients, 92inequality, 36

tight, 36wavelet tight, 37, 46

framelets, 37

Holder spaces, 7, 82Holder-Zygmund spaces, 79

frame characterization, 81Haar system, 38

invariant neighborhood matrix, 13

limit function, 10linear regression, 91

mask, 12mesh, 10

semi-regular, 10moments, 26

2D, 29regular, 26shifts/rescale, 27

multi-resolution analysis, 37

optimal Holder-Zygmund exponent, 90

partition of unity, 16polynomial generation, 16, 21polynomial reproduction, 16, 66projection kernels, 44

radial basis functions, 94Buhmann, 95, 99polyharmonic, 96, 100

ratio test, 91refinement equation, 17

Fourier side, 20resolution level, 11, 37Riesz basis, 44, 48

scaling functions, 40stationary subdivision operator, 9

106

Index

subdivision matrix, 10, 12subdivision schemes, 9

convergence, 10smoothness, 11

sum rules, 14symbol, 20

product, 21

vanishing moments, 37, 46, 80

wavelets, 37

Zygmund class, 8, 86

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